Wikiversity:Colloquium

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

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

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

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

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

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

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

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

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

== Server switch - Your wiki will be read-only for a short time soon ==
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[[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC)
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Hello, I’m Danny from [[:en:w:Wikimedia_Deutschland|Wikimedia Deutschland’s]] [[m:Wikidata_For_Wikimedia_Projects|Wikidata For Wikimedia Projects]] team.&lt;br/&gt;
We have been investigating and examining how Wikidata’s use in the [[Special:RecentChanges]] and [[Special:Watchlist]] pages can be made to be more useful, or less ‘noisy’ (fewer edits that have little value to the reader).

We are currently exploring to make a change in the volume and the type of Wikidata edits that appear in these pages, and '''we need your feedback''' to help us ensure this is a positive change for the Wikis and won’t disrupt your workflows.&lt;br/&gt;
Language links are also known as sitelinks or interlanguage links and describe when a Wikidata item is connected to a Wiki, adding it to the language dropdown tool for other connected Wikis.
[[File:Types of Lang Link changes.png|x350px|frame|A variety of &quot;Language Link&quot; edits as seen in the Recent Changes feed]]

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Further information about this change: [[m:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|m:Wikidata_For_Wikimedia_Projects/Hide_Language_Link_edits]]

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

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

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

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

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

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

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

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

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

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

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

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

To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here].
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;༄ᨒ𖠰 Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 06:22, 18 October 2026 (UTC)

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

== Interface administrators policy==

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

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

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

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

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

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

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

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

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

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 
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'''You will be able to read, but not edit, all wikis for a short period of time.'''

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

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

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

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

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

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

Any thoughts?

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

== New vote for approval of inactivity policy at its talk page ==
I started a vote on approving the inactivity policy, please vote at [[Wikiversity talk:Inactivity policy#Approval]] instead of here. Thank you. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:05, 6 November 2026 (UTC)</text>
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Artificial intelligence involves intelligence that is created through computers or other means other than humanness. This area is for learning, teaching, and research related to artificial intelligence. AI usage has evolved into [[agentic harnesses]] and more. 

==Learning resources==
* [[Artificial neural network]]
* [[Cross-domain AI topics]]

===Sub-pages ===
* [[/Introduction/]]
* [[Artificial intelligence/prospects|Prospects for artificial intelligence]]
** [[Supersymmetry/String Theory based Artificial Intelligence]]
* [[/Lecture aid/]]
* [[/AI content engine economics/]]
* [[/Checking AI output with a second agent/]]
* [[/Giving an AI agent access safely/]]
* [[/Checking an open-source AI project before you depend on it/]]
* [[/Standing instructions for an AI coding agent/]]
* [[/Giving an AI agent a task it can check/]]
* [[/Confident wrong answers from language models/]]
* [[/Checking an AI supervisor's weekly report/]]

==Discussion questions and essay ideas==
* Can artificial intelligence (AI) be used as a tool for good? List applications scenarios of AI. Discuss ethical aspects that are related to decision making and decision support with AI. Define requirements and constraints of AI application that you could derive from your analysis.
* Identify areas of application where AI that can assist staff members e.g. in health care facilities to get more time for social interaction and support for patients? What happens if AI applications fail and patients get harmed? What level of security must be applied and who is responsible for that. Apply the underlying concept on other topics like autonomous driving, ...)
* Analyse the concept of [[machine learning]] and analysis of big data! What is the importance (PRO and CONs) of AI in this context in comparision to other mathematical or computer science methodologies?
* Assume AI is used for pattern recognition in social media data or in general in the context of [[Commercial Data Harvesting]]. Provide examples and analyse the possible consequence of false-positive resp. false negative classification of individuals! What would be your response and what would you suggest to improve the considered system?
* How can the human brain or human mind be emulated using computers? Is it desirable to? 
* What other ethical issues are involved with artificial intelligence? 
* How can artificial intelligence be utilized and created in an ethical way?

=== More Discussion questions and essay ideas===
* Compare prompt engineering to loop engineering to graph engineering. 
** Compare and contrast these different modalities. 
** When and how can each one be useful? 
* What are some different notable agentic harnesses and what the pros and cons of some of the major ones? 

==Educational/research/learning prompt ideas==
* Create pages for an educational wiki that demonstrate how agentic AI can be utilized to build out an educational wiki.
* What goals should an educational wiki have. That is, for a wiki meant for learning, teaching, and research, how should such collaborative content actually benefit the readers and collaborators. 
* How could AI generated video be utilized for learning, teaching, and research? 
* What are economic aspects of universities that readers and creators on distributed  educational wikis can possibly learn from or benefit from? 
* How those who utilize agentic AI benefit from agentic AI utilize decentralized wikis to have their agent collaborate on their behalf and on behalf of humanity? 
* How could Hermes agents collaborate to advance educational wikis - wikis that are meant for learning, teaching, and research? 

=== Wikipedia ===
==== Wikipedia Readings/Pages ====
{{colbegin|4}}
: {{w|Explainable artificial intelligence}}
: {{w|Artificial intelligence}}
: {{w|Accelerating change}}
: {{w|Quantum computing}}
: {{w|Artificial brain}}
: {{w|Artificial general intelligence}}
: {{w|Mind uploading}}
: {{w|Technological singularity}}
: {{w|Intelligence explosion}}
: {{w|Superintelligence}}
: {{w|AI takeover}}
: {{w|Existential risk from artificial general intelligence}}
: {{w|Friendly artificial intelligence}}
: {{w|Computronium}}
: {{w|Distributed artificial intelligence}}
: {{w|Machine learning}}
: {{w|Philosophy of artificial intelligence}}
: {{w|Hardware for artificial intelligence}}
: {{w|Weak AI}}
: {{w|Deep learning}}
: {{w|History of artificial intelligence}}
: {{w|Synthetic intelligence}}
: {{w|Template:Natural language processing}} (template)
: {{w|List of datasets for machine-learning research}}
: {{w|Fine-tuning (machine learning)}}
: {{w|Convolutional neural network}}
: {{w|Natural language processing}}
: {{w|Transformer (machine learning model)}}
: {{w|Prompt engineering}}
: {{w|PaLM}}
: {{w|GPT-4}}
: {{w|Auto-GPT}}
: {{w|BLOOM (language model)}}
: {{w|Hugging Face}}
: {{w|Hallucination (artificial intelligence)}}
: {{w|AI safety}}
: {{w|AI alignment}}
: {{w|Existential risk from artificial general intelligence}}
: {{w|Template:Existential risk from artificial intelligence}} (template)
: {{w|Ethics of artificial intelligence}}
: {{w|Generative artificial intelligence}}
: {{w|Artificial general intelligence}}
: {{w|Artificial intelligence art}}
: {{w|Music and artificial intelligence}}
: {{w|Generative pre-trained transformer}}
: {{w|Regulation of artificial intelligence}}
: {{w|Wikipedia:Large language models}} (policy)
: {{w|Language model}}
: {{w|LLaMA}}
{{colend}}

===== Wikipedia Categories =====
: {{wc|Artificial intelligence}}
:: {{wc|Artificial neural networks}}
::: {{wc|Neural network architectures}}
:: {{wc|Machine learning}}
::: {{wc|Datasets in machine learning|... datasets}}
::: {{wc|Deep learning}}
:::: {{wc|Deep learning software|... software}}
:: {{wc|Generative artificial intelligence|Generative AI}}
:: {{wc|Applications of artificial intelligence}}
::: {{wc|Large language models}}
::: {{wc|Natural language processing}}
::: {{wc|Speech recognition}}
:: {{wc|Existential risk from artificial general intelligence}}
:: [[w:List of artificial intelligence projects|AI projects]]

===External ===
* [https://www.youtube.com/watch?v=Jy3zjXK4ao4 Google Just Put an A.I. Brain in a Robot (Research Breakthrough)] (video)

==See also==
[[Image:Artificial Intelligence &amp; AI &amp; Machine Learning - 30212411048.jpg|thumb|300px|right]]
{{colbegin|3}}
* [[Robotics Projects]]
* [[Introduction to Robotics]]
* [[Arduino Robot]]
* [[Automation]]
* [[History of artificial intelligence]]
* [[Anthropomorphic Robotics Project]]
* [[Large language models]]
* [[Robot car]]
* [[Automatic Control and Robotics]]
* [[Open-source domestic robots]]
* [[Portal:Artificial intelligence]]
* [[Center_for_Corporate_Auditing,_Responsibility_and_Management_policy_Authoring/Artificial_intelligence_eschewal_policy|Artificial intelligence eschewal policy]]
* [[Wikiversity:Artificial_intelligence]] (Policy)
{{colend}}

==External links==
* [https://www.elementsofai.com/ Course on Elements of AI]
* [https://www.reddit.com/r/artificial/ Discuss artificial intelligence on Reddit]
* https://tldr.tech/api/latest/ai
[[Category:Artificial intelligence| ]]</text>
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      <comment>Fixed amibiguous definition (it's usually defined as the estimator of the effect size, not the population effect size itself). Also, discussion of eta squared was a distraction, and calling Cohen's d &quot;appropriate&quot; was editorializing and controversial. Some would argue Hedges g is more appropriate, for example (since Cohen's d is a biased estimator).</comment>
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{{mathematics}}
{{statistics}}
'''Cohen's ''d''''' is a statistic that estimates [[effect size]]. Specifically, it estimates the standardised difference between two means. It can be used, for example, to accompany reporting of [[t-test|''t''-test]] and [[ANOVA]] results. It is also widely used in [[meta-analysis]].

Cohen's ''d'' can be calculated as the difference between the sample means divided by the pooled sample SD: {{center top}}&lt;math&gt;\frac {\text{mean  difference}} {\text{standard deviation}} &lt;/math&gt; 
or &lt;math&gt;\frac {{M_2  -  M_1}} {\text{pooled standard deviation}} &lt;/math&gt;{{center bottom}}


Cohen's ''d'', etc. is available in [[SPSS]] in the t test procedure and a number of other places.

In an ANOVA, you need to be clear about which two means you are interested in knowing about the size of difference between. This could most likely mean that you are interested in several ''d''s, e.g., to compare marginal totals (for main effects) or cells (for interactions). In general, it is recommended to report all relevant Cohen's ''d'' values unless you've got a particular reason to just focus on a one or some of the possible values. From a descriptive statistics table, calculating Cohen's ''d'' is relatively straightforward.

Calculating Cohen's ''d'' provides useful information for discussion (e.g., allows ready comparison with meta-analyses and the size of effects reported in other studies). Where you are reporting about differences between two means, then a standardised mean effect size (such as ''d'') would be an appropriate accompaniment to [[statistical significance|inferential testing]].

==See also==
* [[w:Effect size#Cohen's d|Cohen's ''d'']] (Wikipedia)

==External links==
* [https://www.uccs.edu/lbecker/ Becker's effect size calculator] - an online calculator for Cohen's ''d'' and ''r''
* [http://wilderdom.com/301/Cohensd.xls Cohensd.xls] - an Excel spreadsheet for calculating Cohen's ''d''
* [https://rpsychologist.com/d3/nhst/ Understanding Statistical Power and Significance Testing] - interactive visualization&lt;noinclude&gt;
[[Category:Effect size]]
[[Category:Statistics]]
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      <text bytes="16168" sha1="mh4y3noabimgk8g45pzeuhhae2nd7r2" xml:space="preserve">9 March
Glad to see your log Catarina! I look forward to the resource that you will create on intersectionality. 
[[Special:Contributions/130.243.100.240|130.243.100.240]] 09:45, 9 March 2009 (UTC)Sangeeta

Nugget hgtu byjnh
KOIIA 2, vt -09
Catarina Schmidt

'''Unni Wikan, Generous Betrayal – Politics of Culture in the New Europe'''

Unni Wikan was born in 1944 and is a professor of social anthropology at the University of Oslo (Wikipedia, 2009-03-01). Wikan leads in her book ”Generous Betrayal” a discussion about culture, cultural rights, human rights, identity and democracy. In the centre of the discussion are young women whose parents have immigrated to Norway or Sweden. Cultural rights are put against human rights when those young women are forced to marry, are denied human democratic rights or in the worst scenario are victims of violence and/or murder. Within anthropology the concept of culture has always being used but not always questioned. Different cultures have been described and taken for granted without being mirrored towards laws and conventions about democracy and human rights signed by societies and nations all over the world.  At the same time we are, according to Wikan, building a system of society with traditions and customs containing oppression which stands in open conflict with fundamental human rights. Wikan means therefore that the word culture can be a trap where children and weaker members of a group tend to be losers. The word culture tends to be a label standing for the exotic and strange – which in the name of culture is not being questioned. What those young women in Norway and Sweden, which have been exposed with honor of violence, have in common is that they belong to a patriotic culture associated with traditions which can result in violence and oppression in order to defense the honor of the family. Within this culture there is no room for individualism – variations in aspects of gender, age, ethnicity, social class and education are being denied or reduced. There are simply not many alternatives in being a woman or a man. Since identity is created by gender, age, ethnicity, social class and education the conclusion is a denial of the identity itself. The concept of culture becomes a trap which denies any variations of identity when we instead need the opposite;”a society worth living in must make room for people to craft identities” (p. 74). This sentence from Wikan opens in my opinion up for a meeting between human beings and if the individuals can meet can perhaps in the long run cultures can meet. Because of this Wikan claims that the respect for the individual”may require that”culture” be demoted or resisted” (p. 77). Wikan go as far that she speaks of the concept of culture as”a new concept of race” (p. 79) meaning that it maintains a politic which denies human rights. The author means that since culture is only a concept it has no own autonomic or material existence. Wikan means that culture and power walk hand in hand and that the authority always is resting by those who have the power:

 ''”Some people have the right – or seize the right – to define what is to count and for what, and the result, the authoritative”truth” is often called culture.” (p. 87).''

By exaggerating external and internal differences between different cultures and also to prevent differences within a certain culture a false image of a homogeny culture appears, something that will affect those members of the group that from different reasons want to go other ways than the head line. Wikan argues that the concept of culture”covers up the complexity of human existence, the fact that we are both children of”our culture” and unique individuals” (p. 88). All together I understand that Wikan’s recapitulation about the problems regarding the culture concept can be summarized like this:

*Culture is a concept – not a thing
*There is no objective definition
*Culture has no agency – only humans have the power to act
*Culture has no power – only what people give it
*Cultural things change
*Culture and power go hand in hand – some people seize the right to tell what is the “truth”
*Culture is usually taken to mean difference, not difference and similarity
*There are internal differences but culture tends to play down them which gives a false picture of homogeneity

(after Wikan, p. 88)

The question ringing in my head after having read Wikan’s book is why are we not doing anything? For me the answer is that we very often are blind mice, but with the help of Wikan we can be at least near sighted. 

'''Homi K. Bhaba, The Location of Culture'''

Homi K. Bhabha, an Indian theorist of post-colonialism, was born 1949 and is according to Wikipedia (2009-03-01) currently teaching at Harvard University. From what I understand Bhabha in his book “Location of Culture” consequently punctures thoughts that support ideas about any better or worse of cultures. There is, according to the author a great need to redefine the concept of national cultures:

&lt;blockquote&gt;
The very concept of homogenous national cultures, the 
Consensual or contiguous transmission of historical traditions, 
or ‘organic’ ethnic communities – as the ground of cultural
comparativism – are in a profound process of redefinition. The 
hideous extremity of Serbian nationalism proves that the very 
idea of a pure, ‘ethnically cleansed’ national identity can only 
be achieved through the death, literal and figurative, of the 
complex interweavings of history, and the culturally
contingent borderlines of modern nationhood. (p. 7)
&lt;/blockquote&gt;

His vision includes instead an articulated, clear hybrid of cultures and it is this space that he calls “the third space of enunciation” (p. 53). Within this space Bhabha means that there is a possibility to create a conception of culture that is not based on exotic descriptions which in a one-sided and uninterruptedly way put the others aside as the worse. We can, as I understand Bhabha, achieve this by finding the words where we can talk about ourselves and the others. To ascribe some cultures a kind of intrinsic inherited pureness which is not questionable is, according to Bhabha, untenable. Generations of authors have described meetings between cultures in the old colonies. After having read this book I understand why Bhabha includes so many literary quotations since those give descriptions that cover everything from integration to vast gulfs and oppressing violence between cultures. The Western compulsion to colonize has, of course, had a huge impact on our view of foreign cultures and through colonial and postcolonial authors we can see the possibility to change and, again, the vision of the third space. The different contributions from post colonialism expose both the colonists and the former colonies and have an ambition to explain today’s problems in society as a consequence of the western European colonization. The author’s contribution, as I understand it, is about the cultural and social effects within the post colonialism and by emphasizing the colonists and the colonized dependence from each other he argues that all cultural systems should be constructed in “the third space of enunciation” (p. 53). This, Bhabha means, will help us to escape from “we” and “them” where we are using and giving each other troublesome binaries. Binary oppositions are fatal since they unthinkingly dominate the second, again there is a better and a worse culture and examples might be civilized/savage, rational/irrational, enlightened/ignorant and eventually west/east and white/black. The binaries need according to Bhabha to be destabilized and he argues that culture must be understood to interact in a much more complex manner than those traditional binary oppositions can allow. 

&lt;blockquote&gt;
To take this perspective would mean that we see ‘racism’ 
not simply as a hangover from archaic conceptions of the 
aristocracy, but as part of the historical traditions of civic 
and liberal humanism that create ideological matrices of 
national aspiration, together with the concept of ‘a people’ 
and its imagined community. (p. 359)
&lt;/blockquote&gt;

The idea of race is present through Bhabas book and as I understand it is something we have to deal with and revise in our time. Bhabas uses, as always, the literature and one example is Fanon from “The fact of blackness”:

&lt;blockquote&gt;
‘Dirty nigger!’ Or, simply, Look a Negro’ (p.338)
&lt;/blockquote&gt;

From this quotation Bhabha draws conclusions and the senses of not only being the individual ‘nigger’ but also of belonging to the marginalized group of black people –but also about developing black identity in human with ideological matrices. And shortly afterwards he again quotes Fanon with the sentence:

&lt;blockquote&gt;
You come too late much too late, there will always be a world – 
a whole white world between you and us. (p. 339)
&lt;/blockquote&gt;

Where is culture located and how do we see upon culture through the lenses of race? After having read this book I am more aware of the complexity than before but still very confused. Bhabas book is not an easily read book. The language is intense and compact with many adjectives and many, many quotations from the world’s literature describing colonialism, colonists and the aftermaths from it. The author requires a great deal from its reader and I wonder if he could have done it a little easier and hopefully a bit clearer. His theory is also a controversial one and full of demands. It is about you and me, how we look upon ourselves and others. What he wants is this:

&lt;blockquote&gt;
What is crucial to such a vision of the future is the belief 
that we must not merely change the narratives of our 
histories, but transform our sense of what it means to live,
to be, in other times and different spaces, both human and
historical. (p. 367)
&lt;/blockquote&gt;

'''Philip Riley, Language, Culture and Identity'''

Philip Riley is emeritus professor of Ethnolinguistics at Nancy University in France. A central theme in Rileys book is communicative practices where language plays the most important role and he claims himself that the thrust of the book is “to study the linguistic and communicative aspects of the social knowledge system” (p. 8). From the very start Riley emphasizes the complexity of language, referring to Herder:

&lt;blockquote&gt;
…since language is at one and the same time the tool, the
content and the form of human thought, and every act of
knowledge is only possible through the medium of language.
This interdependence means that the modes of thought and
cultures of people can only be studied and analysed in and 
through their language. (p.9)
&lt;/blockquote&gt;

Rileys book is a cross-disciplinary contribution showing how social knowledge systems and language work to shape and perform individual and collective identities. In almost every chapter he does a historical review and he cites prominent academics all over the book. In his first chapter he lays the foundation and understanding of the concept of identity and explains that it is socially constructed but also a part of the surrounding society. Riley claims that “the reflection on our interactive experiences – facilitated and canalized by language – enables to become who we are” (p. 16). Society itself, language and interaction are, according to Riley, the very sources of individual identity. In chapter two Riley describes the social knowledge system and the linking and primary mechanism is, again, language. Another crucial ingredient is intersubjectivity which can be explained as a state of shared meaning between the participants and seen by Riley as necessary for the development of identity:

&lt;blockquote&gt;
The ability to establish intersubjectivity, to enter into social
and meaningful contact with another, is a necessary condition
for the formation of identity. (p. 33)
&lt;/blockquote&gt;

In this way communication and culture are dependent from one another. Riley argues, as Wikan, that the term culture is polysemic, but in a more pragmatic and humoristic way:

&lt;blockquote&gt;
Individuals acquire and construct their personal cultural repertoires
on the basis of the interactional opportunities available to them.  
A country-dweller and a town-dweller, a doctor’s child and a
labourer’s child, a Muslim and a Catholic will have access to
participation in different ranges of interaction……In other words,
acquiring culture is not like serving oneself from a soup tureen, 
where every bowlful has identical contents, it is more like lunching
from a vast smorgasbord. (p. 39)
&lt;/blockquote&gt;

Culture continues to exist through language and Riley seems to mean that culture in itself is knowledge. This knowledge is distributed on different levels; the family level, shared by all in the neighbourhood and then shared by everyone on another level. Riley argues that each kind of knowledge corresponds to an aspect of social identities like family and communities like professions, religious groups or sport clubs. This is for me very close to the concept of literacy – the ability to read and write is always connected with social activities and domains which include relationships and a history in a chronological order. Through all these activities is the competence of communications used and Riley points out how different competences of language are needed, with in my opinion, an outstanding metaphor:

&lt;blockquote&gt;
…linguistic competence means being a grammarian, 
communicative competence means being a speaker and
sociocultural competence means being a member. Rather,
the three can be imagined like Russian dolls, nesting inside
another, so that sociocultural competence includes communicative
competence which in turn, includes linguistic competence. (p. 53)
&lt;/blockquote&gt;

So, the knowledge of culture is communicated in different communicative practices and for this are the competences of language cited above needed. According to Riley communicative practices do not only communicate messages, they also communicate social identity which “is made up of a configuration of memberships and each membership is knowledge-and-language based” (p. 113). And, again, the conclusion is, that if we want to study identity we have to study language. Identity is formed by subject-positions materialized of power relationships and Riley gives a lot of example of this beginning with the gendarme with his uniform and revolver shouting at an innocent boy who in that second becomes a criminal. Communicative practices can also be gendered when we say things like “big boys don’t cry” (p. 139) and “nice girls don’t use words like that” (p. 139). 

Riley also dwells over the concept of the stranger, quoting Simmel and Schütz and with the help of the later Riley points out different kinds of mobility addressing membership and belonging. One example is the candidate who wishes to become a member of an exclusive club and another is the new recruit joining the army. In order to form social identity recognition is needed – the stranger has to be recognized by others. Riley refers to Hegel and says:

&lt;blockquote&gt;
It is otherness which makes interaction both possible and
Necessary. I can only exist as an individual if other 
individuals exist and if we know it. A human community is
a community of recognition: it can only exist to the extent
that the individuals concerned recognize the existence of one
another. (p.176)
&lt;/blockquote&gt;

The Me is, according to Riley, seen as the individual’s own perception of his or her social identity while the memberships of social groups form a bundle with roles and positions which the Self occupies. Eventually public perceptions of the Me and the Self form ethos. For me Riley has tried to answer the question about where identity is and how it is created. The answer is a of course complex but also as simple as his last words “just as history tells us who we are, identity is made of the stories we tell ourselves” (p. 244).</text>
      <sha1>mh4y3noabimgk8g45pzeuhhae2nd7r2</sha1>
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    <revision>
      <id>2844731</id>
      <parentid>2844718</parentid>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/Artjomtouchedme|Artjomtouchedme]] ([[User_talk:Artjomtouchedme|talk]]) to last version by [[User:MaintenanceBot|MaintenanceBot]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="16159" sha1="2kbu124es6vyeta47xjoaiovyhh0f14" xml:space="preserve">9 March
Glad to see your log Catarina! I look forward to the resource that you will create on intersectionality. 
[[Special:Contributions/130.243.100.240|130.243.100.240]] 09:45, 9 March 2009 (UTC)Sangeeta

Läslogg
KOIIA 2, vt -09
Catarina Schmidt

'''Unni Wikan, Generous Betrayal – Politics of Culture in the New Europe'''

Unni Wikan was born in 1944 and is a professor of social anthropology at the University of Oslo (Wikipedia, 2009-03-01). Wikan leads in her book ”Generous Betrayal” a discussion about culture, cultural rights, human rights, identity and democracy. In the centre of the discussion are young women whose parents have immigrated to Norway or Sweden. Cultural rights are put against human rights when those young women are forced to marry, are denied human democratic rights or in the worst scenario are victims of violence and/or murder. Within anthropology the concept of culture has always being used but not always questioned. Different cultures have been described and taken for granted without being mirrored towards laws and conventions about democracy and human rights signed by societies and nations all over the world.  At the same time we are, according to Wikan, building a system of society with traditions and customs containing oppression which stands in open conflict with fundamental human rights. Wikan means therefore that the word culture can be a trap where children and weaker members of a group tend to be losers. The word culture tends to be a label standing for the exotic and strange – which in the name of culture is not being questioned. What those young women in Norway and Sweden, which have been exposed with honor of violence, have in common is that they belong to a patriotic culture associated with traditions which can result in violence and oppression in order to defense the honor of the family. Within this culture there is no room for individualism – variations in aspects of gender, age, ethnicity, social class and education are being denied or reduced. There are simply not many alternatives in being a woman or a man. Since identity is created by gender, age, ethnicity, social class and education the conclusion is a denial of the identity itself. The concept of culture becomes a trap which denies any variations of identity when we instead need the opposite;”a society worth living in must make room for people to craft identities” (p. 74). This sentence from Wikan opens in my opinion up for a meeting between human beings and if the individuals can meet can perhaps in the long run cultures can meet. Because of this Wikan claims that the respect for the individual”may require that”culture” be demoted or resisted” (p. 77). Wikan go as far that she speaks of the concept of culture as”a new concept of race” (p. 79) meaning that it maintains a politic which denies human rights. The author means that since culture is only a concept it has no own autonomic or material existence. Wikan means that culture and power walk hand in hand and that the authority always is resting by those who have the power:

 ''”Some people have the right – or seize the right – to define what is to count and for what, and the result, the authoritative”truth” is often called culture.” (p. 87).''

By exaggerating external and internal differences between different cultures and also to prevent differences within a certain culture a false image of a homogeny culture appears, something that will affect those members of the group that from different reasons want to go other ways than the head line. Wikan argues that the concept of culture”covers up the complexity of human existence, the fact that we are both children of”our culture” and unique individuals” (p. 88). All together I understand that Wikan’s recapitulation about the problems regarding the culture concept can be summarized like this:

*Culture is a concept – not a thing
*There is no objective definition
*Culture has no agency – only humans have the power to act
*Culture has no power – only what people give it
*Cultural things change
*Culture and power go hand in hand – some people seize the right to tell what is the “truth”
*Culture is usually taken to mean difference, not difference and similarity
*There are internal differences but culture tends to play down them which gives a false picture of homogeneity

(after Wikan, p. 88)

The question ringing in my head after having read Wikan’s book is why are we not doing anything? For me the answer is that we very often are blind mice, but with the help of Wikan we can be at least near sighted. 

'''Homi K. Bhaba, The Location of Culture'''

Homi K. Bhabha, an Indian theorist of post-colonialism, was born 1949 and is according to Wikipedia (2009-03-01) currently teaching at Harvard University. From what I understand Bhabha in his book “Location of Culture” consequently punctures thoughts that support ideas about any better or worse of cultures. There is, according to the author a great need to redefine the concept of national cultures:

&lt;blockquote&gt;
The very concept of homogenous national cultures, the 
Consensual or contiguous transmission of historical traditions, 
or ‘organic’ ethnic communities – as the ground of cultural
comparativism – are in a profound process of redefinition. The 
hideous extremity of Serbian nationalism proves that the very 
idea of a pure, ‘ethnically cleansed’ national identity can only 
be achieved through the death, literal and figurative, of the 
complex interweavings of history, and the culturally
contingent borderlines of modern nationhood. (p. 7)
&lt;/blockquote&gt;

His vision includes instead an articulated, clear hybrid of cultures and it is this space that he calls “the third space of enunciation” (p. 53). Within this space Bhabha means that there is a possibility to create a conception of culture that is not based on exotic descriptions which in a one-sided and uninterruptedly way put the others aside as the worse. We can, as I understand Bhabha, achieve this by finding the words where we can talk about ourselves and the others. To ascribe some cultures a kind of intrinsic inherited pureness which is not questionable is, according to Bhabha, untenable. Generations of authors have described meetings between cultures in the old colonies. After having read this book I understand why Bhabha includes so many literary quotations since those give descriptions that cover everything from integration to vast gulfs and oppressing violence between cultures. The Western compulsion to colonize has, of course, had a huge impact on our view of foreign cultures and through colonial and postcolonial authors we can see the possibility to change and, again, the vision of the third space. The different contributions from post colonialism expose both the colonists and the former colonies and have an ambition to explain today’s problems in society as a consequence of the western European colonization. The author’s contribution, as I understand it, is about the cultural and social effects within the post colonialism and by emphasizing the colonists and the colonized dependence from each other he argues that all cultural systems should be constructed in “the third space of enunciation” (p. 53). This, Bhabha means, will help us to escape from “we” and “them” where we are using and giving each other troublesome binaries. Binary oppositions are fatal since they unthinkingly dominate the second, again there is a better and a worse culture and examples might be civilized/savage, rational/irrational, enlightened/ignorant and eventually west/east and white/black. The binaries need according to Bhabha to be destabilized and he argues that culture must be understood to interact in a much more complex manner than those traditional binary oppositions can allow. 

&lt;blockquote&gt;
To take this perspective would mean that we see ‘racism’ 
not simply as a hangover from archaic conceptions of the 
aristocracy, but as part of the historical traditions of civic 
and liberal humanism that create ideological matrices of 
national aspiration, together with the concept of ‘a people’ 
and its imagined community. (p. 359)
&lt;/blockquote&gt;

The idea of race is present through Bhabas book and as I understand it is something we have to deal with and revise in our time. Bhabas uses, as always, the literature and one example is Fanon from “The fact of blackness”:

&lt;blockquote&gt;
‘Dirty nigger!’ Or, simply, Look a Negro’ (p.338)
&lt;/blockquote&gt;

From this quotation Bhabha draws conclusions and the senses of not only being the individual ‘nigger’ but also of belonging to the marginalized group of black people –but also about developing black identity in human with ideological matrices. And shortly afterwards he again quotes Fanon with the sentence:

&lt;blockquote&gt;
You come too late much too late, there will always be a world – 
a whole white world between you and us. (p. 339)
&lt;/blockquote&gt;

Where is culture located and how do we see upon culture through the lenses of race? After having read this book I am more aware of the complexity than before but still very confused. Bhabas book is not an easily read book. The language is intense and compact with many adjectives and many, many quotations from the world’s literature describing colonialism, colonists and the aftermaths from it. The author requires a great deal from its reader and I wonder if he could have done it a little easier and hopefully a bit clearer. His theory is also a controversial one and full of demands. It is about you and me, how we look upon ourselves and others. What he wants is this:

&lt;blockquote&gt;
What is crucial to such a vision of the future is the belief 
that we must not merely change the narratives of our 
histories, but transform our sense of what it means to live,
to be, in other times and different spaces, both human and
historical. (p. 367)
&lt;/blockquote&gt;

'''Philip Riley, Language, Culture and Identity'''

Philip Riley is emeritus professor of Ethnolinguistics at Nancy University in France. A central theme in Rileys book is communicative practices where language plays the most important role and he claims himself that the thrust of the book is “to study the linguistic and communicative aspects of the social knowledge system” (p. 8). From the very start Riley emphasizes the complexity of language, referring to Herder:

&lt;blockquote&gt;
…since language is at one and the same time the tool, the
content and the form of human thought, and every act of
knowledge is only possible through the medium of language.
This interdependence means that the modes of thought and
cultures of people can only be studied and analysed in and 
through their language. (p.9)
&lt;/blockquote&gt;

Rileys book is a cross-disciplinary contribution showing how social knowledge systems and language work to shape and perform individual and collective identities. In almost every chapter he does a historical review and he cites prominent academics all over the book. In his first chapter he lays the foundation and understanding of the concept of identity and explains that it is socially constructed but also a part of the surrounding society. Riley claims that “the reflection on our interactive experiences – facilitated and canalized by language – enables to become who we are” (p. 16). Society itself, language and interaction are, according to Riley, the very sources of individual identity. In chapter two Riley describes the social knowledge system and the linking and primary mechanism is, again, language. Another crucial ingredient is intersubjectivity which can be explained as a state of shared meaning between the participants and seen by Riley as necessary for the development of identity:

&lt;blockquote&gt;
The ability to establish intersubjectivity, to enter into social
and meaningful contact with another, is a necessary condition
for the formation of identity. (p. 33)
&lt;/blockquote&gt;

In this way communication and culture are dependent from one another. Riley argues, as Wikan, that the term culture is polysemic, but in a more pragmatic and humoristic way:

&lt;blockquote&gt;
Individuals acquire and construct their personal cultural repertoires
on the basis of the interactional opportunities available to them.  
A country-dweller and a town-dweller, a doctor’s child and a
labourer’s child, a Muslim and a Catholic will have access to
participation in different ranges of interaction……In other words,
acquiring culture is not like serving oneself from a soup tureen, 
where every bowlful has identical contents, it is more like lunching
from a vast smorgasbord. (p. 39)
&lt;/blockquote&gt;

Culture continues to exist through language and Riley seems to mean that culture in itself is knowledge. This knowledge is distributed on different levels; the family level, shared by all in the neighbourhood and then shared by everyone on another level. Riley argues that each kind of knowledge corresponds to an aspect of social identities like family and communities like professions, religious groups or sport clubs. This is for me very close to the concept of literacy – the ability to read and write is always connected with social activities and domains which include relationships and a history in a chronological order. Through all these activities is the competence of communications used and Riley points out how different competences of language are needed, with in my opinion, an outstanding metaphor:

&lt;blockquote&gt;
…linguistic competence means being a grammarian, 
communicative competence means being a speaker and
sociocultural competence means being a member. Rather,
the three can be imagined like Russian dolls, nesting inside
another, so that sociocultural competence includes communicative
competence which in turn, includes linguistic competence. (p. 53)
&lt;/blockquote&gt;

So, the knowledge of culture is communicated in different communicative practices and for this are the competences of language cited above needed. According to Riley communicative practices do not only communicate messages, they also communicate social identity which “is made up of a configuration of memberships and each membership is knowledge-and-language based” (p. 113). And, again, the conclusion is, that if we want to study identity we have to study language. Identity is formed by subject-positions materialized of power relationships and Riley gives a lot of example of this beginning with the gendarme with his uniform and revolver shouting at an innocent boy who in that second becomes a criminal. Communicative practices can also be gendered when we say things like “big boys don’t cry” (p. 139) and “nice girls don’t use words like that” (p. 139). 

Riley also dwells over the concept of the stranger, quoting Simmel and Schütz and with the help of the later Riley points out different kinds of mobility addressing membership and belonging. One example is the candidate who wishes to become a member of an exclusive club and another is the new recruit joining the army. In order to form social identity recognition is needed – the stranger has to be recognized by others. Riley refers to Hegel and says:

&lt;blockquote&gt;
It is otherness which makes interaction both possible and
Necessary. I can only exist as an individual if other 
individuals exist and if we know it. A human community is
a community of recognition: it can only exist to the extent
that the individuals concerned recognize the existence of one
another. (p.176)
&lt;/blockquote&gt;

The Me is, according to Riley, seen as the individual’s own perception of his or her social identity while the memberships of social groups form a bundle with roles and positions which the Self occupies. Eventually public perceptions of the Me and the Self form ethos. For me Riley has tried to answer the question about where identity is and how it is created. The answer is a of course complex but also as simple as his last words “just as history tells us who we are, identity is made of the stories we tell ourselves” (p. 244).</text>
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  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
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      <timestamp>2026-11-07T11:01:10Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2844803</origin>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261106.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261106.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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    <revision>
      <id>2844805</id>
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      <timestamp>2026-11-07T11:02:11Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261107.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261106.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261107.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261107.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

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  <page>
    <title>Atheism</title>
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      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-58926-46|~2026-58926-46]] ([[User_talk:~2026-58926-46|talk]]) to last version by [[User:Hasley|Hasley]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="599" sha1="qq00f1b9t0bbltg0cshteezf12mhv5r" xml:space="preserve">[[File:Atheismsymbol_endorsed_by_AAI.svg|thumb|120x120px|Symbol designed by Atheist Alliance International in 2007]]
Atheism is, in a broad sense, the rejection of belief in the existence of deities. In a narrower sense, atheism is specifically the position that there are no deities. Most inclusively, atheism is the absence of belief that any deities exist. Atheism is contrasted with theism, which, in its most general form, is the belief that at least one deity exists.&lt;ref&gt;[[Wikipedia: Atheism]]&lt;/ref&gt;

== Readings ==
* [[Wikipedia: Atheism]]

== References ==
{{Reflist}}

[[Category:Atheism]]</text>
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    <title>World Languages/South America</title>
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      <text bytes="3121" sha1="ons7gnmhnmdyq2wctxjxs0fen681flq" xml:space="preserve">[[File:Map of South America.png|thumb|right|400px|Map of South America]]
'''South America''' includes 14 countries: Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Falkland Guyana, Paraguay, Peru, Suriname, Uruguay and Venezuela. Spanish is the official language of all South American countries except Brazil, Guyana, Suriname and French Guiana, and is spoken even in countries that are not historically Spanish. Portuguese is the official language of Brazil. Official languages in Guyana, Suriname and French Guiana are respectively English, Dutch and French.

Before colonization from European nations, South America was inhabited by several Aboriginal natives speaking a wide variety of languages from different language families, most of them being long forgotten now, but a lot are still known by few and are currently in danger of extinction. At the time of European contact, it is estimated that 1,500 languages were spoken in South America; only 350 of those languages are still spoken today. Quechua is the native language family with the most speakers.

The classification and studies of indigenous languages in South America is not very advanced compared to the classification of North American indigenous languages. As such, it is difficult to determine what languages are related to each other to develop proper  families.

== List of indigenous languages ==
[[File:SouthAmerican families 03.png|thumb|right|400px|Extensive language families (more than 5 languages) of South America; dark spots are language isolates or quasi-isolates and grey spots are unclassified languages; Queucha, the language family with the most speakers, is not shown]]
''This list only includes languages with living speakers.''

* Akawaio (Brazil, Guyana, Venezuela)
* Arawak (French Guinea, Guyana, Suriname, Venezuela)
** Achagua
** Alto Perene
** Amuesha
** Apuriña
** Asháninka/Ashéninka
** Baniva
** Barawana
** Baure
** Cabiyari
** Caquinte
** Chamicuro
** Enawene Nawe
** Iñapari
** Kinikinau
** Kurripako
** Lokono Dian
** Machiguenga
** Maliwan
** Marawa
** Mehináku
** Moxos: Trinitario, Ignaciano
** Nomatisguenga
** Palikúr
** Paraujano
** Paresí
** Piro
** Resígaro
** Saraveka
** [[/Taíno/]] (revitalization efforts)
** Tariana
** Terena
** Wapishana
** Waraikú
** Warekena
** Waurá
** Wayuunaíki
** Yavitero
** Yawalapiti
* Arutani (Brazil, Venezuela)
* Barí (Colombia, Venezuela)
* Carib (French Guinea, Guyana, Suriname, Venezuela)
** Amonap
** Apalaí
** Arára
** Akuriyó
** Atruahí
** Bacairí
** Carijona
** Cumanagoto
** Galibi
** Hianákoto
** Hixkaryána
** Japréria
** Kapóng
** Macushi
** Mapoyo
** Opon
** Panare
** Pemon
** Salumá
** Tamanaku
** Tiriyó
** Txikão
** Waiwai
** Wayana
** Ye'kuana
** Yukpa
* Chaima (Venezuela)
* Cuiba (Colombia, Venezuela)
* Cumanagoto (Venezuela)
* E’ñapa Woromaipu (Venezuela)
* ''To be completed...''

== Other languages ==
* German, Colonia Tovar (Venezuela)
* ''To be completed...''

== Extinct languages ==
* Baniva
* ''To be completed...''

{{center top}}[[Portal:Wikilang]]{{center bottom}}

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    <title>Universal Bibliography/Countries</title>
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      <text bytes="43412" sha1="nwj2qi8ifq5po001qfj1roj2c33k8nm" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Geography|Geography]].

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

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

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

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

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

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

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Encyclopedias

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

Biographical dictionaries
*Wispelway. Japanese Biographical Index [https://books.google.co.uk/books?id=4UkhAAAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Seiichi Iwao. Biographical Dictionary of Japanese History. Kodansha International. 1978.[https://books.google.co.uk/books?id=TkQOAQAAMAAJ]
*Senʼichi Hisamatsu. Biographical Dictionary of Japanese Literature. Kodansha International. 1976. [https://books.google.com/books?id=X3lkAAAAMAAJ]
*Yutaka Tazawa (ed). Biographical Dictionary of Japanese Art. Kodansha International. 1981. [https://books.google.com/books?id=7ozrAAAAMAAJ]
*Japan Biographical Encyclopedia &amp; Who's Who. 1958. Review: [https://books.google.co.uk/books?id=lV_VAAAAMAAJ 25] Contemporary Japan 280
**Japan Biographical Encyclopedia &amp; Who's Who (2nd Edition) 1961. The Japan Biographical Research Department, The Rengo Press Ltd. Preface dated October 1960. Commentary: [https://books.google.co.uk/books?id=-SZSAAAAMAAJ]
***1961 Supplement. December 1961.
*Shunjiro Kurita. Who's Who in Japan. [https://archive.org/details/WhosWhoInJapan1913/page/n24/mode/1up 2nd Ed]. Catalogue: [https://cir.nii.ac.jp/crid/1971712334741056302]. Review: New York Times, 3 October 1915 [https://www.nytimes.com/1915/10/03/archives/japans-whos-who-printed-in-english-new-volume-shows-rapid-changes.html].
**[[w:ja:石川安次郎|Yasujiro Ishikawa]] (ed). Fifth Annual Edition. The Who's Who in Japan Publishing Department: The Keiseisha. No 15, 2-chome, Owari-cho, Ginza, Kyobashi-ku, Tokyo. Preface dated May 1916. [https://books.google.co.uk/books?id=joBJAAAAYAAJ]. Yasujiro Ishikawa's biography &quot;Ishikawa Yasujiro&quot; is on p 213.

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

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

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

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

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

Summaries of the press
*Daily Summary of Japanese Press

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

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

Newspapers

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

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

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

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

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

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

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

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

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

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

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

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

[Series]
*Routledge Contemporary Japan Series

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

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

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

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

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

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

Languages

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

Literature

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

Music

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

Cinema

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

Television

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

Architecture

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

Culture

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

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

Tokyo
*J S Eades. Tokyo. (World Bibliographical Series, vol 214). CLIO Press. 1999. ISBN 1-85109-292-7.

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

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

[[Category:Countries]]</text>
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      <text bytes="43463" sha1="3xpr8k72nzzw3rzjmt3afldyerf50dq" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Geography|Geography]].

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

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

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

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

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

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

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Encyclopedias

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

Biographical dictionaries
*Wispelway. Japanese Biographical Index [https://books.google.co.uk/books?id=4UkhAAAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Seiichi Iwao. Biographical Dictionary of Japanese History. Kodansha International. 1978.[https://books.google.co.uk/books?id=TkQOAQAAMAAJ]
*Senʼichi Hisamatsu. Biographical Dictionary of Japanese Literature. Kodansha International. 1976. [https://books.google.com/books?id=X3lkAAAAMAAJ]
*Yutaka Tazawa (ed). Biographical Dictionary of Japanese Art. Kodansha International. 1981. [https://books.google.com/books?id=7ozrAAAAMAAJ]
*Japan Biographical Encyclopedia &amp; Who's Who. 1958. Review: [https://books.google.co.uk/books?id=lV_VAAAAMAAJ 25] Contemporary Japan 280
**Japan Biographical Encyclopedia &amp; Who's Who (2nd Edition) 1961. The Japan Biographical Research Department, The Rengo Press Ltd. Preface dated October 1960. Commentary: [https://books.google.co.uk/books?id=-SZSAAAAMAAJ]
***1961 Supplement. December 1961.
*Shunjiro Kurita. Who's Who in Japan. [https://archive.org/details/WhosWhoInJapan1913/page/n24/mode/1up 2nd Ed]. Catalogue: [https://cir.nii.ac.jp/crid/1971712334741056302]. Review: New York Times, 3 October 1915 [https://www.nytimes.com/1915/10/03/archives/japans-whos-who-printed-in-english-new-volume-shows-rapid-changes.html].
**[[w:ja:石川安次郎|Yasujiro Ishikawa]] (ed). Fifth Annual Edition. The Who's Who in Japan Publishing Department: The Keiseisha. No 15, 2-chome, Owari-cho, Ginza, Kyobashi-ku, Tokyo. Preface dated May 1916. [https://books.google.co.uk/books?id=joBJAAAAYAAJ]. Yasujiro Ishikawa's biography &quot;Ishikawa Yasujiro&quot; is on p 213.

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

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

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

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

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

Summaries of the press
*Daily Summary of Japanese Press

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

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

Newspapers

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

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

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

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

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

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

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

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

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

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

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

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

[Series]
*Routledge Contemporary Japan Series

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

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

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

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

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

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

Languages

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

Literature

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

Music

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

Cinema

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

Television

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

Architecture

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

Culture

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

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

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

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

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

[[Category:Countries]]</text>
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  <page>
    <title>Virasoro CFTs with a large central charge</title>
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      <comment>Mazac formalism</comment>
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      <text bytes="8851" sha1="p229skagbo12b3a4pb080vj0wwb0d1c" xml:space="preserve">From the [[w:AdS/CFT correspondence]], we expect the existence of two-dimensional CFTs with a large central charge. In order to be dual to three-dimensional gravity, such CFTs are also expected to have only Virasoro symmetry, a vacuum state with a large gap above it, and to be non-diagonal and unitary. 

== Motivations from AdS/CFT and three-dimensional gravity ==

Summarize main CFT implications of the works of Fitzpatrick, Kaplan, Walters.&lt;ref name=&quot;fkw14&quot;/&gt;&lt;ref name=&quot;fkw15&quot;/&gt;

:&lt;math&gt; c= \frac{3\ell_\text{AdS}}{2G_\text{Newton}}&lt;/math&gt;

Role of chaos.

Role of the light, heavy limits.

Matter vs pure gravity.

== Predicted properties == 

== Known two-dimensional CFTs ==

Here we review some known two-dimensional CFTs whose chiral symmetry algebras are the Virasoro algebra. None of them has all the desired properties of a CFT dual to three-dimensional gravity.

=== Solvable non-diagonal CFTs ===

In the &lt;math&gt;Q&lt;/math&gt;-state Potts model and the &lt;math&gt;O(n)&lt;/math&gt; model are non-diagonal CFTs with discrete spectrums.
Their non-diagonal fields have dimensions &lt;math&gt;(\Delta,\bar\Delta)=(\Delta_{(r,s)},\Delta_{(r,-s)})&lt;/math&gt; with &lt;math&gt;r,s\in\mathbb{Q}&lt;/math&gt;. Such dimensions, with fractional Kac indices, are natural because the condition for the spin to be integer boils down to &lt;math&gt;rs\in\mathbb{Z}&lt;/math&gt;, and because integer Kac indices play an important role in the solvability of [[w:minimal model (physics)|minimal models]], [[w:Liouville theory]], etc. But the total conformal dimension is then 
:&lt;math&gt; \Delta_{(r,s)}+\Delta_{(r,-s)} = \frac{c-1}{12} - \frac12 r^2 b^2 -\frac12 s^2 b^{-2} \quad \text{with} \quad c = 13 + 6b^2 + 6 b^{-2}
&lt;/math&gt;
This is bounded from below for &lt;math&gt; r,s\to \infty&lt;/math&gt; provided&lt;ref name=&quot;gnj21&quot;/&gt; 
:&lt;math&gt;
\Re b^2 &lt; 0  \iff \Re c &lt; 13
&lt;/math&gt;
So CFTs of this type cannot exist for &lt;math&gt;c\to +\infty&lt;/math&gt;. They are also non-unitary, and the identity may or may not be the ground state, depending on the central charge.

There is an analogous obstruction in integrable approaches to conformal blocks. The relation between [[w:Virasoro conformal blocks]] at &lt;math&gt;c=1&lt;/math&gt; and solutions of the Painlevé VI equation involves an infinite sum over blocks. The natural generalization of this sum at generic central charge would diverge for &lt;math&gt;\Re c&gt;13&lt;/math&gt;.

=== Liouville theory ===

[[w:Liouville theory]] is unitary and exists for any &lt;math&gt;c\in\mathbb{C}&lt;/math&gt;, but it is diagonal, does not have a vacuum state in the spectrum, and there is no gap above the ground state.

=== Generalized minimal models ===

[[w:Minimal_model_(physics)#Generalized_minimal_models|Generalized minimal models]] have discrete spectrums made of degenerate fields, and exist for any &lt;math&gt;c\in\mathbb{C}&lt;/math&gt;. However, the spectrum is not bounded from below if &lt;math&gt;\Re c&gt;13&lt;/math&gt;. This does not prevent GMMs from existing, thanks to their fusion rules.

== Possible approaches ==

=== Understanding analytic solutions of the crossing symmetry equations ===

In order to better understand the analytic properties of the fusing matrix, and of the crossing symmetry equations themselves, it would be interesting to study the known analytic solutions. In quite a few cases, analytic solutions are known, but it has not been proved analytically that they are in fact solutions. 

In simple diagonal CFTs such as [[w:Liouville theory]] and diagonal [[w:minimal model (physics)|minimal models]], the 3pt structure constant is a special case of the fusing matrix, and crossing symmetry follows from the Pentagon equation.&lt;ref name=&quot;rib14&quot;/&gt; There is a similar story about D-series minimal models.&lt;ref name=&quot;run99&quot;/&gt; Other analytic solutions are probably less well understood. 

=== Solving the crossing symmetry equations at c=25 ===

At &lt;math&gt;c=1&lt;/math&gt;, the crossing symmetry equations are more explicit and easier to solve than at generic &lt;math&gt;c&lt;/math&gt;. This is due to the relation of [[w:Virasoro conformal block#Solutions_of_the_Painlevé_VI_equation|conformal blocks]] with solutions of the Painlevé VI equation and isomonodromic connections. This leads to an explicit formula for the fusing matrix that is much simpler than the general formula.&lt;ref name=&quot;ilt13&quot;/&gt; As a result, one can try to solve crossing symmetry equations analytically.&lt;ref name=&quot;gs18&quot;/&gt;

The cases &lt;math&gt;c=25&lt;/math&gt; and &lt;math&gt;c=1&lt;/math&gt; are related by &lt;math&gt;b\to ib&lt;/math&gt; and are formally very similar. This raises the hope that simple formulas for the fusing matrix can also be found at &lt;math&gt;c=25&lt;/math&gt;. Actually, the known general Ponsot-Teschner formula for the fusing matrix is valid for &lt;math&gt;c\in \mathbb{C}-(-\infty,1)&lt;/math&gt;, the general formula for &lt;math&gt;c\leq 1&lt;/math&gt; is still missing, and finding it raises similar issues. Knowing that Liouville theory is crossing-symmetric is a big hint.

=== Analyticity in spin ===

To find manageable solutions, we should use the right variables. At small central charge, the right variables seem to be Kac indices and conformal blocks, which reflect the existence of degenerate fields. At large central charge, degenerate fields might not play an important role. But if we are not using Kac indices, how do we efficiently encode the structure of a non-diagonal spectrum, and the integer values of the conformal spin? 

Maybe we should use the analyticity in spin properties that are known to occur in CFTs with global symmetry. Functions of the integer spin would be traded for analytic functions of a continuous variable. In particular, conformal blocks would be traded for Regge blocks.&lt;ref name=&quot;chs20&quot;/&gt; This might allow us to write non-diagonal spectrums that are bounded from below, which cannot easily be done with Kac indices if &lt;math&gt;\Re c&gt;13&lt;/math&gt;.

=== Noncommutative geometry ===

A formulation of 2d CFT in terms of noncommutative geometry was proposed by Mazac,&lt;ref name=&quot;maz26&quot;/&gt; motivated by compact unitary CFTs with &lt;math&gt;c\geq 1&lt;/math&gt;, although it applies more generally. 

== References ==

{{Reflist|refs=
&lt;ref name=&quot;maz26&quot;&gt;{{cite web | last=Mazac | first=Dalimil | title=Conformal field theory from automorphic forms on a quantum group | website=arXiv.org | date=2026-11-04 | url=https://arxiv.org/abs/2611.06907 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;chs20&quot;&gt;{{cite web | last=Caron-Huot | first=Simon | last2=Sandor | first2=Joshua | title=Conformal Regge Theory at Finite Boost | website=arXiv.org | date=2020-08-26 | doi=10.1007/JHEP05(2021)059 | url=https://arxiv.org/abs/2008.11759v1 | access-date=2021-11-23}}&lt;/ref&gt;
&lt;ref name=&quot;run99&quot;&gt;{{cite web | last=Runkel | first=Ingo | title=Structure constants for the D-series Virasoro minimal models | website=arXiv.org | date=1999-08-05 | doi=10.1016/S0550-3213(99)00707-5 | url=https://arxiv.org/abs/hep-th/9908046v2 | access-date=2021-11-08}}&lt;/ref&gt;
&lt;ref name=&quot;rib14&quot;&gt;{{cite web | last=Ribault | first=Sylvain | title=Conformal field theory on the plane | website=arXiv.org | date=2014-06-17 | url=https://arxiv.org/abs/1406.4290v5 | access-date=2021-11-08}}&lt;/ref&gt;
&lt;ref name=&quot;gs18&quot;&gt;{{cite web | last=Gavrylenko | first=Pavlo | last2=Santachiara | first2=Raoul | title=Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions | website=arXiv.org | date=2018-12-26 | doi=10.1007/JHEP11(2019)119 | url=https://arxiv.org/abs/1812.10362v3 | access-date=2021-11-08}}&lt;/ref&gt;
&lt;ref name=&quot;ilt13&quot;&gt;{{cite web | last=Iorgov | first=N. | last2=Lisovyy | first2=O. | last3=Tykhyy | first3=Yu. | title=Painlev\'e VI connection problem and monodromy of c=1 conformal blocks | website=arXiv.org | date=2013-08-19 | doi=10.1007/JHEP12(2013)029 | url=https://arxiv.org/abs/1308.4092v2 | access-date=2021-11-08}}&lt;/ref&gt;
&lt;ref name=&quot;fkw15&quot;&gt;{{cite web | last=Fitzpatrick | first=A. Liam | last2=Kaplan | first2=Jared | last3=Walters | first3=Matthew T. | title=Virasoro Conformal Blocks and Thermality from Classical Background Fields | website=arXiv.org | date=2015-01-21 | doi=10.1007/JHEP11(2015)200 | url=https://arxiv.org/abs/1501.05315v3 | access-date=2021-11-05}}&lt;/ref&gt;
&lt;ref name=&quot;fkw14&quot;&gt;{{cite web | last=Fitzpatrick | first=A. Liam | last2=Kaplan | first2=Jared | last3=Walters | first3=Matthew T. | title=Universality of Long-Distance AdS Physics from the CFT Bootstrap | website=arXiv.org | date=2014-03-26 | doi=10.1007/JHEP08(2014)145 | url=https://arxiv.org/abs/1403.6829v2 | access-date=2021-11-05}}&lt;/ref&gt;
&lt;ref name=&quot;gnj21&quot;&gt;{{cite web | last=Grans-Samuelsson | first=Linnea | last2=Nivesvivat | first2=Rongvoram | last3=Jacobsen | first3=Jesper Lykke | last4=Ribault | first4=Sylvain | last5=Saleur | first5=Hubert | title=Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model | website=arXiv.org | date=2021-11-01 | url=https://arxiv.org/abs/2111.01106v1 | access-date=2021-11-04}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
      <sha1>p229skagbo12b3a4pb080vj0wwb0d1c</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2844810</id>
      <parentid>2844628</parentid>
      <timestamp>2026-11-07T11:12:45Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2844810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="c0df0m4ek57nlvyshp5a92tv3ct0ths" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261105.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>c0df0m4ek57nlvyshp5a92tv3ct0ths</sha1>
    </revision>
    <revision>
      <id>2844812</id>
      <parentid>2844810</parentid>
      <timestamp>2026-11-07T11:13:38Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2844812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="7a5iu3j4la9w3wf4n073wzca8j18fw3" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261106.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>7a5iu3j4la9w3wf4n073wzca8j18fw3</sha1>
    </revision>
    <revision>
      <id>2844814</id>
      <parentid>2844812</parentid>
      <timestamp>2026-11-07T11:14:50Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2844814</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="irqckky8cvejs7cozkonfdpfze5elty" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261107.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>irqckky8cvejs7cozkonfdpfze5elty</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Maia.sh</title>
    <ns>3</ns>
    <id>289175</id>
    <revision>
      <id>2844725</id>
      <parentid>2441216</parentid>
      <timestamp>2026-11-06T16:47:43Z</timestamp>
      <contributor>
        <username>Mfield</username>
        <id>573392</id>
      </contributor>
      <minor />
      <comment>Mfield moved page [[User talk:Dudhhr]] to [[User talk:Maia.sh]]: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/Dudhhr|Dudhhr]]&quot; to &quot;[[Special:CentralAuth/Maia.sh|Maia.sh]]&quot;</comment>
      <origin>2441216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2637" sha1="7727dgku5pdkog9mpw65awu5v31yc40" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Dudhhr!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
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* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
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* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
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&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:25, 20 October 2022 (UTC)&lt;/div&gt;
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    <title>User:Dc.samizdat/Real Euclidean four-dimensional space</title>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120467" sha1="7xgekapwynyz5v9v033os66ld87jknn" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique set of rotations. The geometry of each rotation's isocline is that of a star polygon. The relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space, the stretch to imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>7xgekapwynyz5v9v033os66ld87jknn</sha1>
    </revision>
    <revision>
      <id>2844739</id>
      <parentid>2844738</parentid>
      <timestamp>2026-11-06T19:35:48Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120496" sha1="byae45qmxbfzmh83l28563yt49rv5nr" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space, the stretch to imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844741</id>
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      <timestamp>2026-11-06T19:47:22Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120524" sha1="ahh441rgn6ih9coenls3nfosaxc3t6g" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ahh441rgn6ih9coenls3nfosaxc3t6g</sha1>
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    <revision>
      <id>2844742</id>
      <parentid>2844741</parentid>
      <timestamp>2026-11-06T19:54:02Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120572" sha1="jl4ihdmll4h595o34wtnrsruipyt6bj" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>jl4ihdmll4h595o34wtnrsruipyt6bj</sha1>
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    <revision>
      <id>2844743</id>
      <parentid>2844742</parentid>
      <timestamp>2026-11-06T19:57:50Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120602" sha1="l8tjhyyhaui0tk7g3svgg3ejjx8xtij" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>l8tjhyyhaui0tk7g3svgg3ejjx8xtij</sha1>
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    <revision>
      <id>2844744</id>
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      <timestamp>2026-11-06T20:06:34Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120621" sha1="ma2fnpbjjc37vqllaha1ilu9mcd5cnk" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ma2fnpbjjc37vqllaha1ilu9mcd5cnk</sha1>
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    <revision>
      <id>2844745</id>
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      <timestamp>2026-11-06T20:14:08Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120678" sha1="lb7vhi4wx9z6f13nkxhqr095cpx6vql" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844746</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120685" sha1="ceik9ggedw7n6glpmv647o7hwtn7h30" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ceik9ggedw7n6glpmv647o7hwtn7h30</sha1>
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    <revision>
      <id>2844747</id>
      <parentid>2844746</parentid>
      <timestamp>2026-11-06T20:18:40Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120685" sha1="amsp9x23ikn4chntr1t1v3tpx4e1fdq" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844748</id>
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      <timestamp>2026-11-06T20:22:57Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120708" sha1="862pmpfim8z0tsxog9wbqwu1n4oior7" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If you view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>862pmpfim8z0tsxog9wbqwu1n4oior7</sha1>
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    <revision>
      <id>2844749</id>
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      <timestamp>2026-11-06T20:26:34Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120722" sha1="0tuh1u7uu86khspr4adfizd5fjvefow" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If you view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, you can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>0tuh1u7uu86khspr4adfizd5fjvefow</sha1>
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    <revision>
      <id>2844750</id>
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      <timestamp>2026-11-06T20:27:19Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120720" sha1="nmyvk54cykplwxfw7jrzizhn73r0oc3" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we can find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>nmyvk54cykplwxfw7jrzizhn73r0oc3</sha1>
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    <revision>
      <id>2844751</id>
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      <timestamp>2026-11-06T20:27:55Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120716" sha1="d6r1yhnnsfzmtpi00271xrwj152mv30" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>d6r1yhnnsfzmtpi00271xrwj152mv30</sha1>
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    <revision>
      <id>2844752</id>
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      <timestamp>2026-11-06T20:29:32Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120721" sha1="cwz4lloh9amwm1rsdcxds3nf9guhrtl" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>cwz4lloh9amwm1rsdcxds3nf9guhrtl</sha1>
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    <revision>
      <id>2844753</id>
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      <timestamp>2026-11-06T20:30:44Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120726" sha1="mrylka8u33etzkwoubzaqy65hr2lp0k" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>mrylka8u33etzkwoubzaqy65hr2lp0k</sha1>
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    <revision>
      <id>2844769</id>
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      <timestamp>2026-11-06T22:24:29Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="120729" sha1="nxisv6hp8ui76wgdn2oiqunjfnco8k7" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is intelligence.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2844770</id>
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        <username>Dc.samizdat</username>
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      <text bytes="120728" sha1="jn9ympd5i2ecmn706hjlahfrv5veq39" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere whose center is wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844771</id>
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        <username>Dc.samizdat</username>
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      <text bytes="120721" sha1="e2itnp4iebwj7mbjtafjia66m6ikwhh" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844773</id>
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        <username>Dc.samizdat</username>
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      <text bytes="120760" sha1="jjy6k4id00anlzccinfgxw8oieros4o" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between successive star polygons are distinctly mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>jjy6k4id00anlzccinfgxw8oieros4o</sha1>
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    <revision>
      <id>2844777</id>
      <parentid>2844773</parentid>
      <timestamp>2026-11-06T22:50:08Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120792" sha1="kbcf9dtptjq8t0suko71hvhwbdl19fs" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>kbcf9dtptjq8t0suko71hvhwbdl19fs</sha1>
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    <revision>
      <id>2844783</id>
      <parentid>2844777</parentid>
      <timestamp>2026-11-06T22:58:03Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120805" sha1="qdr21p0k1inakvnb9zv5ag7t52wfl1p" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. In 4-space inertia is the conservation of this distinct double rotation. In 4-space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844786</id>
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      <timestamp>2026-11-06T23:02:13Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120787" sha1="9rryheoq9nmfiqak5h6ulxlm2qv6zzt" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's conservation law, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. It is a challenge to visualize this double rotation.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>9rryheoq9nmfiqak5h6ulxlm2qv6zzt</sha1>
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    <revision>
      <id>2844789</id>
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      <timestamp>2026-11-06T23:07:04Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* General relativity is the geometry of gravity in Euclidean 4-space */</comment>
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      <text bytes="120787" sha1="r06zuiuk2y1ujhxrkpam0hb9xi982ln" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>r06zuiuk2y1ujhxrkpam0hb9xi982ln</sha1>
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    <revision>
      <id>2844791</id>
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      <timestamp>2026-11-06T23:41:30Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="121198" sha1="gyqo2n5xx0lvh7530vkog35conf8iog" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atom of the element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations are potential nuclides of the atomic element, some potentially more than others.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>gyqo2n5xx0lvh7530vkog35conf8iog</sha1>
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    <revision>
      <id>2844792</id>
      <parentid>2844791</parentid>
      <timestamp>2026-11-06T23:47:26Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="121258" sha1="8bdg7myexgnusawrtc4wqzu914cqabk" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atom of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations are potential nuclides of the atomic element, some potentially more than others.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>8bdg7myexgnusawrtc4wqzu914cqabk</sha1>
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    <revision>
      <id>2844793</id>
      <parentid>2844792</parentid>
      <timestamp>2026-11-06T23:50:19Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="121270" sha1="klm6t3utzkysdji3h8pi37oc6srsp2l" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atom of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations are potential nuclides of the atomic element, some potentially more than others.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

.... We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2844795</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="122729" sha1="2xgnaozswxi85w539n96v0zyleqm0nd" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - October 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the 4-simplex polytope. The neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a &lt;math&gt;4&lt;/math&gt;-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:
&lt;blockquote&gt;''Time ticks at the same rate everywhere in space.''&lt;/blockquote&gt;
This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space. That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. .... We conclude that the rotational geometry of the proton is that of the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number.  The rotational symmetry of each element is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atom of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations are potential nuclides of the atomic element, some potentially more than others.

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. The 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the orthonormal basis polytope, whose size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integral number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Additionally, the largest 4-polytopes in the series, the 120-point 600-cell and the 600-point 120-cell, are each divisible into an integral number of non-disjoint 4-orthoplexes which share vertices, in several different ways. While it is impossible to position two 4-simplexes on the 3-sphere such that they share vertices (except by making them coincident), two 4-orthoplexes can be positioned so that they share an axis, or even two axes (a central plane), without sharing all 8 vertices.

An element of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. Thus for example the element Carbon with 6 protons occupies 30 distinct vertices of the 600. One of these configurations is also a configuration of an integral number of disjoint 4-orthoplexes, but the rest are not.

 ....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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  <page>
    <title>WikiJournal Preprints/Valid publication of names proposed in &quot;A new genomic taxonomy system for the Synechococcus collective&quot;</title>
    <ns>0</ns>
    <id>327105</id>
    <revision>
      <id>2844817</id>
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      <timestamp>2026-11-07T11:42:50Z</timestamp>
      <contributor>
        <username>Artoria2e5</username>
        <id>705354</id>
      </contributor>
      <comment>10.1111/1462-2920.15173</comment>
      <origin>2844817</origin>
      <model>wikitext</model>
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      <text bytes="2934" sha1="lrwhog8im5htqczwtoyky9ptou7rfbl" xml:space="preserve">{{Article info
|journal         = WikiJournal of Science      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
|last1           = Wang
|first1          = Mingye
|last2           =
|first2          =
|last3           =
|first3          =
|last4           =
|first4          = &lt;!-- up to 9 authors can be added in this above format --&gt;
|et_al           = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
|affiliations    = Anthon Open Source Community
|correspondence  = arthur200126[@]gmail.com
|keywords        = Cyanobacteria, taxonomy, Synechococcus, valid publication, new taxa
|license         = &lt;!-- default is CC-BY --&gt;
|abstract        =

A comprehensive taxonomic classification of the Synechococcus collective (SC) [10.1111/1462-2920.15173] was proposed in 2020 by Salazar et al.,
but the names were not validly published. This article constitutes valid publication of Cyanobacteriales ord. nov.,
Leptococcales ord. nov., Thermosynechococcales ord. nov., Phormidesmiales ord. nov., Limnotrichaceae fam. nov.,
Leptococcaceae fam. nov, Brevicoccaceae fam. nov, Phormidesmidaceae fam. nov., ''Lacustricoccus'' gen. nov,
''Synechospongium'' gen. nov under the rules of the International Code of Nomenclature for Algae, Fungi, and Plants (ICN).

}}

== Introduction ==
Cyanobacteria of the genus ''[[Synechococcus]]'' s.l. are among the most widespread and abundant photosynthetic microorganisms on Earth, inhabiting marine, freshwater, and thermal environments. Molecular studies have revealed significant diversity within the ''Synechococcus'' collective (SC), leading to proposals for a more refined taxonomic classification. In 2020, Salazar et al. proposed a comprehensive taxonomic framework for the SC based on genomic data based on the methods of the Genome Taxonomy Database, but the names were not validly published according to the International Code of Nomenclature for Algae, Fungi, and Plants (ICN). This article aims to provide valid publication for several new taxa within the SC, including new orders, families, and genera.

== Methods ==
A list of proposed taxa was compiled from Table 1 of Salazar et al. (2020). The taxonomic names were checked against the
List of Prokaryotic names with Standing in Nomenclature (LPSN) to ensure that they were not previously published. Spelling
corrections were taken from LPSN where applicable. Descriptions for each taxon were derived from the original publication, with additional details provided as necessary to meet the requirements for valid publication.

== Results and Discussion ==


==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

==References==
{{reflist|35em}}

[[Category:WikiJournal]]
[[Category:Taxonomy]]</text>
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    <title>Wikiversity:Inactivity policy</title>
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      <text bytes="1216" sha1="s6jqr82s6yc0hz9uyuy7sto14e6l7di" xml:space="preserve">{{Proposal|WV:INACTIVITY}}
Wikiversity [[Wikiversity:Support staff|support staff]] ([[Wikiversity:Curatorship|curators]], [[Wikiversity:Custodianship|custodians]], and [[Wikiversity:Bureaucratship|bureaucrats]]) are considered inactive if, over the previous 12 months, they have made no edits or logged administrative actions.

== Process ==
# A notice is sent to the [[Wikiversity:Colloquium|Colloquium]], listing the inactive support staff.
# Inactive support staff are notified, explaining that their rights may be removed, via their talk page (with {{tlx|Inactivity notice}}) and [[Special:EmailUser]].
# If no on-wiki response is received from the inactive support staff member within a month, the administrative rights will be removed:
#* Curators: A bureaucrat will remove the permission locally.
#* Custodians and bureaucrats: A steward will be asked to remove the permission(s) (per [[m:Steward requests/Permissions#Removal of access]]) along with any other advanced rights.&lt;ref&gt;Per [[m:Requests for comment/Removal of advanced rights]] on Meta.&lt;/ref&gt;

== Note ==
{{reflist}}

==See also==
* [[meta:Admin activity review/Local inactivity policies]]

[[Category:Wikiversity administration|Inactivity policy]]</text>
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      <id>2844727</id>
      <parentid>2843107</parentid>
      <timestamp>2026-11-06T17:03:56Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Approval */ +</comment>
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      <text bytes="12996" sha1="4fynrfgvuinkx08v7mupgkr6k4j6gn3" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)
: I think this practice helps towards openness and transparency. For example, someone else in the community may become aware and comment about reasons why the SSM may be inactive. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:43, 21 July 2026 (UTC)
::I concur that the community should be made aware of, and given an opportunity to comment on, decisions that have an effect on our site. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:11, 27 August 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)
: I think 12 months is a simple, clear, appropriate timeframe for inactivity.
: I agree that one month to reply is more reasonable than 2 weeks. It is quite common for people to be on annual leave e.g., for 4 weeks. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:47, 21 July 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)
::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) &lt;em&gt;or&lt;/em&gt; blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:26, 16 July 2026 (UTC)
:::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC)
::: Yes, I agree with Koavf's suggestion; it resolves Mu301's concerns that Bureaucrat-specific actions may not be used in a 12 month period even though the user is active. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:50, 21 July 2026 (UTC)
:::: Koavf's suggestion adequately addresses my concerns. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:15, 27 August 2026 (UTC)
:: The current draft refers to administrative actions in a broad sense, not bureaucrat-specific actions. So, I think it is reasonable to interpret this as meaning that a bureaucrat would only be considered inactive if they haven't used any administrative action and haven't edited in 12 months. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:31, 21 July 2026 (UTC)
::: I concur with that interpretation. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:44, 7 October 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
:--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)
: For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC)
: In that case, go ahead. I am not against these changes. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC)
:: Pinging @[[User:Mu301|Mu301]] a second time–in case you did not see my response(s). [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:03, 23 August 2026 (UTC)

==Vested interest==
Which admins could be affected by this proposal? I think that we should ping them to solicit input on a proposed policy before finalizing the terms. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:43, 28 August 2026 (UTC)
: User:Evolution and evolvability. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 18:46, 7 October 2026 (UTC)

== Approval ==
Is this policy good enough, or does it need additional changes? I await feedback here. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 22:52, 24 October 2026 (UTC)
: I think the proposed policy is ready (or near ready) for discussion by the Wikiversity community (e.g., by notifying via sitenotice). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:55, 25 October 2026 (UTC)
:: Instead of the colloquium, we can hold a vote here.
:: We might also need to add a sentence per [[m:Requests for comment/Removal of advanced rights]]; if custodians are desysopped, their advanced permissions are also removed if they have any. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 21:46, 25 October 2026 (UTC)
:::I agree that it is near ready. In general, I think it is better to hold discussion on the policy talk page, rather than colloquium to make archives easier to find. I haven't (yet) reviewed the current wording. --mikeu [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:15, 28 October 2026 (UTC)
:::: I agree that it is generally better to hold policy discussions on policy talk pages because it keeps community deliberations attached to the policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:37, 28 October 2026 (UTC)

=== Votes ===
* {{support}} as proposer, and because after I made some final adjustments and changes, I believe it is ready to become a policy. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:03, 6 November 2026 (UTC)</text>
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    </revision>
    <revision>
      <id>2844729</id>
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      <timestamp>2026-11-06T17:38:49Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
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      <comment>/* Votes */</comment>
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      <text bytes="13268" sha1="m681m8at2ng0omn2sx7ncv130yhntlb" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)
: I think this practice helps towards openness and transparency. For example, someone else in the community may become aware and comment about reasons why the SSM may be inactive. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:43, 21 July 2026 (UTC)
::I concur that the community should be made aware of, and given an opportunity to comment on, decisions that have an effect on our site. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:11, 27 August 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)
: I think 12 months is a simple, clear, appropriate timeframe for inactivity.
: I agree that one month to reply is more reasonable than 2 weeks. It is quite common for people to be on annual leave e.g., for 4 weeks. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:47, 21 July 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)
::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) &lt;em&gt;or&lt;/em&gt; blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:26, 16 July 2026 (UTC)
:::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC)
::: Yes, I agree with Koavf's suggestion; it resolves Mu301's concerns that Bureaucrat-specific actions may not be used in a 12 month period even though the user is active. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:50, 21 July 2026 (UTC)
:::: Koavf's suggestion adequately addresses my concerns. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:15, 27 August 2026 (UTC)
:: The current draft refers to administrative actions in a broad sense, not bureaucrat-specific actions. So, I think it is reasonable to interpret this as meaning that a bureaucrat would only be considered inactive if they haven't used any administrative action and haven't edited in 12 months. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:31, 21 July 2026 (UTC)
::: I concur with that interpretation. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:44, 7 October 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
:--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)
: For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC)
: In that case, go ahead. I am not against these changes. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC)
:: Pinging @[[User:Mu301|Mu301]] a second time–in case you did not see my response(s). [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:03, 23 August 2026 (UTC)

==Vested interest==
Which admins could be affected by this proposal? I think that we should ping them to solicit input on a proposed policy before finalizing the terms. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:43, 28 August 2026 (UTC)
: User:Evolution and evolvability. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 18:46, 7 October 2026 (UTC)

== Approval ==
Is this policy good enough, or does it need additional changes? I await feedback here. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 22:52, 24 October 2026 (UTC)
: I think the proposed policy is ready (or near ready) for discussion by the Wikiversity community (e.g., by notifying via sitenotice). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:55, 25 October 2026 (UTC)
:: Instead of the colloquium, we can hold a vote here.
:: We might also need to add a sentence per [[m:Requests for comment/Removal of advanced rights]]; if custodians are desysopped, their advanced permissions are also removed if they have any. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 21:46, 25 October 2026 (UTC)
:::I agree that it is near ready. In general, I think it is better to hold discussion on the policy talk page, rather than colloquium to make archives easier to find. I haven't (yet) reviewed the current wording. --mikeu [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:15, 28 October 2026 (UTC)
:::: I agree that it is generally better to hold policy discussions on policy talk pages because it keeps community deliberations attached to the policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:37, 28 October 2026 (UTC)

=== Votes ===
* {{support}} as proposer, and because after I made some final adjustments and changes, I believe it is ready to become a policy. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:03, 6 November 2026 (UTC)
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:38, 6 November 2026 (UTC)</text>
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      <comment>/* Votes */ reply: {{support}} (-) ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="13421" sha1="p7vlh09cd6q5uy6h1iw58qkk0hp9hqm" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)
: I think this practice helps towards openness and transparency. For example, someone else in the community may become aware and comment about reasons why the SSM may be inactive. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:43, 21 July 2026 (UTC)
::I concur that the community should be made aware of, and given an opportunity to comment on, decisions that have an effect on our site. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:11, 27 August 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)
: I think 12 months is a simple, clear, appropriate timeframe for inactivity.
: I agree that one month to reply is more reasonable than 2 weeks. It is quite common for people to be on annual leave e.g., for 4 weeks. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:47, 21 July 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)
::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) &lt;em&gt;or&lt;/em&gt; blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:26, 16 July 2026 (UTC)
:::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC)
::: Yes, I agree with Koavf's suggestion; it resolves Mu301's concerns that Bureaucrat-specific actions may not be used in a 12 month period even though the user is active. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:50, 21 July 2026 (UTC)
:::: Koavf's suggestion adequately addresses my concerns. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:15, 27 August 2026 (UTC)
:: The current draft refers to administrative actions in a broad sense, not bureaucrat-specific actions. So, I think it is reasonable to interpret this as meaning that a bureaucrat would only be considered inactive if they haven't used any administrative action and haven't edited in 12 months. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:31, 21 July 2026 (UTC)
::: I concur with that interpretation. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:44, 7 October 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
:--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)
: For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC)
: In that case, go ahead. I am not against these changes. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC)
:: Pinging @[[User:Mu301|Mu301]] a second time–in case you did not see my response(s). [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:03, 23 August 2026 (UTC)

==Vested interest==
Which admins could be affected by this proposal? I think that we should ping them to solicit input on a proposed policy before finalizing the terms. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:43, 28 August 2026 (UTC)
: User:Evolution and evolvability. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 18:46, 7 October 2026 (UTC)

== Approval ==
Is this policy good enough, or does it need additional changes? I await feedback here. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 22:52, 24 October 2026 (UTC)
: I think the proposed policy is ready (or near ready) for discussion by the Wikiversity community (e.g., by notifying via sitenotice). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:55, 25 October 2026 (UTC)
:: Instead of the colloquium, we can hold a vote here.
:: We might also need to add a sentence per [[m:Requests for comment/Removal of advanced rights]]; if custodians are desysopped, their advanced permissions are also removed if they have any. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 21:46, 25 October 2026 (UTC)
:::I agree that it is near ready. In general, I think it is better to hold discussion on the policy talk page, rather than colloquium to make archives easier to find. I haven't (yet) reviewed the current wording. --mikeu [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:15, 28 October 2026 (UTC)
:::: I agree that it is generally better to hold policy discussions on policy talk pages because it keeps community deliberations attached to the policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:37, 28 October 2026 (UTC)

=== Votes ===
* {{support}} as proposer, and because after I made some final adjustments and changes, I believe it is ready to become a policy. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:03, 6 November 2026 (UTC)
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:38, 6 November 2026 (UTC)
* {{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:59, 6 November 2026 (UTC)</text>
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      <comment>#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]</comment>
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      <text bytes="71702" sha1="m28mxtooeqaulfxqsxu8y2f9uygkkkd" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
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{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
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==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
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{{MEBF/2026
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# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* more detailed review of the best psychological research about the topic
#* quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]—express more in your own words
#* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter
&lt;!-- Overall - Edit summaries --&gt;
#* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
#* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
#* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
#* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
#* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
#* This chapter appears to be human-written
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
#* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
#* In some places, there is overreliance on genAI
#* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
#* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
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# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Include a relevant image
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
#  of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* more detailed review of the best psychological research about the topic
#* quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]—express more in your own words
#* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter
&lt;!-- Overall - Edit summaries --&gt;
#* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
#* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
#* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
#* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
#* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
#* This chapter appears to be human-written
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
#* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
#* In some places, there is overreliance on genAI
#* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
#* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in  [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use 
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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      <text bytes="71784" sha1="opabh70uihv9nlg1112cq3uah41520e" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
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~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested) 
&lt;!-- Overall - Edit summaries --&gt;
#* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
#* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
#* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
#* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
#* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
#* This chapter appears to be human-written
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
#* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
#* In some places, there is overreliance on genAI
#* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
#* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Include a relevant image
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
#  of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested) 
&lt;!-- Overall - Edit summaries --&gt;
#* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
#* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
#* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
#* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
#* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
#* This chapter appears to be human-written
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
#* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
#* In some places, there is overreliance on genAI
#* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
#* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
#* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in  [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use 
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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    </revision>
    <revision>
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        <id>10242</id>
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      <format>text/x-wiki</format>
      <text bytes="71790" sha1="eesq1jb4nwqkdr0vst6dtgdsyl5042e" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
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# 
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|5=
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# 
|6=
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# 
|7=
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# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested) 
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Include a relevant image
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
#  of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested) 
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use 
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent/very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
      <sha1>eesq1jb4nwqkdr0vst6dtgdsyl5042e</sha1>
    </revision>
  </page>
  <page>
    <title>The John Snow Prediabetes Institute</title>
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      <text bytes="6689" sha1="j81aa8unpa824lsfs8o2vwjjhg9cfa8" xml:space="preserve">&lt;big&gt;The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)&lt;/big&gt;

&lt;big&gt;'''Background:''' Millions of people are at an increased risk of developing metabolic syndrome, starting with atherosclerosis, pre-diabetes, pre-hypertension, hypertension, and being overweight before the age of 30, often without any symptoms.[1] These conditions constitute the background for developing type 2 diabetes mellitus, cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, young skilled professionals, nurses, and medical students. Alternatively, data from the academic literature can be utilized to identify the high risk group by requesting authors of the publications to re-analyse the data. Stratifying in age-groups is needed as the root causes driving elevated blood sugar look very different depending on the demographic.&lt;/big&gt;  

&lt;big&gt;The 16-weeks '''intervention studies''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt;. The new challenging intervention studies include the AI-Digital lifestyle programs.&lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational material is available from the international diabetes organisations e.g ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|
&lt;big&gt;Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt;

]]'''&lt;big&gt;Prevalence studies&lt;/big&gt;'''  

[[/The Maritime Health database design // |&lt;big&gt;1.1 The Maritime Health database design&lt;/big&gt;]]

&lt;big&gt;1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.5  Excel recoding of the clinical data (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)&lt;/big&gt;

&lt;span lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;big&gt;'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.&lt;/big&gt;&lt;/span&gt;

&lt;big&gt;16 weeks human caused Intervention studies: English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention
&lt;ref&gt;Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.&lt;/ref&gt;&lt;/big&gt;

'''&lt;big&gt;Organisation&lt;/big&gt;'''

&lt;big&gt;- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]&lt;/big&gt;

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

&lt;big&gt;- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;
[[Category:Prediabetes ]]
[[Category:Prediabetes ]]
&lt;big&gt;'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;&lt;/big&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain,  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;&lt;/small&gt; 
==References==</text>
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      <timestamp>2026-11-06T14:25:30Z</timestamp>
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        <username>JSINST</username>
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      <text bytes="6685" sha1="9b1h78df0fn6qp3zpjrsvuf3pckllpg" xml:space="preserve">&lt;big&gt;The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)&lt;/big&gt;

&lt;big&gt;'''Background:''' Millions of people are at an increased risk of developing metabolic syndrome, starting with atherosclerosis, pre-diabetes, pre-hypertension, hypertension, and being overweight before the age of 30, often without symptoms.[1] These conditions constitute the background for developing type 2 diabetes mellitus, cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, young skilled professionals, nurses, and medical students. Alternatively, data from the academic literature can be utilized to identify the high risk group by requesting authors of the publications to re-analyse the data. Stratifying in age-groups is needed as the root causes driving elevated blood sugar look very different depending on the demographic.&lt;/big&gt;  

&lt;big&gt;The 16-weeks '''intervention studies''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt;. The new challenging intervention studies include the AI-Digital lifestyle programs.&lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational material is available from the international diabetes organisations e.g ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|
&lt;big&gt;Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt;

]]'''&lt;big&gt;Prevalence studies&lt;/big&gt;'''  

[[/The Maritime Health database design // |&lt;big&gt;1.1 The Maritime Health database design&lt;/big&gt;]]

&lt;big&gt;1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;1.5  Excel recoding of the clinical data (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;&lt;/big&gt;

&lt;big&gt;'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)&lt;/big&gt;

&lt;span lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;big&gt;'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.&lt;/big&gt;&lt;/span&gt;

&lt;big&gt;16 weeks human caused Intervention studies: English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention
&lt;ref&gt;Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.&lt;/ref&gt;&lt;/big&gt;

'''&lt;big&gt;Organisation&lt;/big&gt;'''

&lt;big&gt;- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]&lt;/big&gt;

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

&lt;big&gt;- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;&lt;/big&gt;
[[Category:Prediabetes ]]
[[Category:Prediabetes ]]
&lt;big&gt;'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;&lt;/big&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain,  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;&lt;/small&gt; 
==References==</text>
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    <title>Motivation and emotion/Book/2026/Extended process model of emotion regulation</title>
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      <text bytes="29684" sha1="nub32vvlp6jkq3op2oewuz4ds4evfgt" xml:space="preserve">{{title|Extended process model of emotion regulation:&lt;br&gt;What is the extended process model and how does it explain the regulation of emotions in different contexts?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=6}}
[[File:School level test of Wikischolar 2026 at Ispahani Public School &amp; College 08.jpg|thumb|'''Figure 1'''. Ethan sitting his stressful exam {{ic|Which one is Ethan?}}]]

''' Case Study - Exam stressors'''

Ethan is halfway through a university exam in a small classroom filled with 30 other examinees. Suddenly, he realises that he has misread a question and wasted the past 15 minutes. His heart rate spikes, his thoughts begin to race, and panic sets in. Overwhelmed by the sudden onset of negativity, he feels the urge to give up.

Normally, Ethan copes with [[W: Psychological stress | stress]] and [[anxiety]] through exercise, as he finds this to be the most consistent modulator of his emotions. However, in an exam room, this strategy is no longer available.

Ethan pauses and recognises that his current negativity is not helping. He identifies that panic is hindering his performance and that emotion regulation is needed if he wants to do well. He begins telling himself, “I can still get partial marks for that question.” Realising that the positive self‑talk is helping, he continues: “For this next question, I’ll make sure to focus on all facets of what it asks.” By the end of the next question, Ethan once again feels a sense of confidence in his ability to pass the exam.
{{RoundBoxBottom}}

Emotion regulation refers to the processes through which individuals monitor, evaluate, and modify their emotional responses. Contemporary psychological science emphasises that regulation is not a single act but a dynamic, context-sensitive system that unfolds over time (Bonanno &amp; Burton, 2013). Gross's (2015) Extended Process Model (EPM) captures this complexity by organising regulation into three iterative stages: identification, selection and implementation. Each is governed by a higher-order valuation system that determines whether, when, and how emotions should be regulated. This framework highlights that regulatory success depends not only on the strategy chosen but on the fit between regulatory goals, contextual affordances, and the individual's cognitive resources.

Emotional regulation is central to mental health. [[Motivation and emotion/Book/2026/Emotion dysregulation|Emotional dysregulation]] predicts a wide range of difficulties, including the strong association between dysregulation and anxiety in autistic youth (Connor et al., 2020). Meta-analytic evidence further demonstrates that habitual reappraisal is linked to better well-being, whereas suppression is often linked to poorer outcomes, though these patterns seem to vary across contexts (Chen et al., 2025). These findings underscore the need to understand ''why'' emotional regulation works differently across individuals and situations.

{{RoundBoxTop|theme=6}}
'''Focus questions'''

* How do cultural [[W: social norms | norms]] shape the emotional regulation process?
* What role does [[W: cognitive flexibility | cognitive flexibility]] play in the EPM’s ability to explain individual differences in regulatory effectiveness?
*Does the EPM explain a difference between children and adults in regulation ability, effectiveness, and strategy selection?
* How does the EPM explain the strong predictive link between [[Motivation and emotion/Book/2026/Emotion dysregulation | emotional dysregulation]] and anxiety in autistic youth?
* Is there an optimal way to emotionally regulate?

{{RoundBoxBottom}}

== The Extended Process Model of Emotional Regulation ==
Emotion regulation can be understood as a higher‑order valuation system that evaluates and modifies first‑order emotional responses. A higher‑order valuation system refers to a meta‑level evaluative process that monitors the outputs of lower‑level emotional systems and determines whether those responses align with one’s goals, contextual demands, and longer‑term priorities {{f}}. As Gross (2015) explains, regulation occurs when such a higher‑order system “takes another valuation system… that is generating emotion as its input, evaluates it either negatively or positively, and activates action impulses that are intended to modify the activity of the first‑level valuation system”. In this view, emotion regulation is not merely a strategy but a goal‑driven process that dynamically shapes unfolding emotional reactions. Within this framework, Gross (2015) proposes a three‑stage regulatory cycle comprising identification, selection, and implementation. These stages operate iteratively, with ongoing monitoring of regulatory effectiveness guiding adjustments across the cycle {{f}}.

=== Identification: ===
The identification stage concerns the fundamental question of whether emotion regulation is required at all. According to Gross (2015), this stage is governed by the higher-order valuation system that evaluates the unfolding emotional response against current goals, contextual affordances, and motivational priorities. Identification involves detecting discrepancies between desired and actual emotional states. Essentially, recognising that the current, experienced emotion is misaligned with what the situation demands {{g}}. This is an evaluative process, integrating prior learning, cultural norms and situational constraints to determine the necessity of regulation {{f}}. Failure at this stage, such as misidentifying an emotion as irrelevant or failing to recognise a [[w: Maladaptation | maladaptive]] trajectory can derailed {{g}} the entire regulatory cycle {{f}}. Thus, the identification stage acts as the gatekeeper of emotional regulation, ensuring cognitive resources are deployed only when emotional change is instrumentally valuable {{f}}.

=== Selection: ===
Once regulation is deemed necessary, the selection stage determines what strategy should be used. Gross (2015) emphasises that this stage involves choosing among diverse regulatory families, such as attentional deployment, cognitive change, or response modulation, based on their anticipated fit with situational demands. Selection is a predictive process; the valuation system simulates the likely consequences of each strategy, weighing cognitive costs, contextual differences in regulatory repertoire (Bonanno &amp; Burton, 2013), meaning that people with broader, more flexible strategy selection requires both contextual sensitivity and strategic foresight; poor selection may lead to strategies that are cognitively overtaxing, situationally inappropriate, or counterproductive. In this way, selection acts as the model's optimisation mechanism, aligning regulatory tactics with the dynamic structure of the emotional episode. {{f}}

=== Implementation: ===
Implementation concerns how the chosen strategy is enacted in real time. Gross (2015) describes this stage as the translation of abstract regulatory intentions into concrete tactics, such as shifting attention, reframing meaning or modulating expressive behavior. Implementation is effortful and dynamic; it unfolds over time, requiring sustained cognitive control and continuous adjustment as the emotional episode evolves. Crucially, implementation is not a single action but an iterative process in which the tactic is maintained, intensified, or abandoned depending on its moment-to-moment effectiveness. Research has shown that ineffective implementation predicts strategy switching (Adamczyk et al., 20204), highlighting that implementation is tightly coupled with feedback processes. Thus, implementation is the engine of regulation, where theoretical strategies become embodied actions that shape the emotional trajectory.

'''Table 1'''.

Stages of EPM {{ic|spell out abbreviation for table caption}} {{f}}

{| class=&quot;wikitable&quot;
|'''Identification'''
|Involves determining whether an emotional response requires regulation by evaluating its relevance to one’s goals and situational demands.
|-
|'''Selection'''
|Choosing the most contextually appropriate and goal‑aligned regulatory strategy from the available options.
|-
|'''Implementation'''
|Enacting the chosen strategy through specific cognitive, behavioural, or attentional mechanisms to modify the unfolding emotional response.
|}

==The Effects of Culture on Emotional Regulation==
Cultural norms play a decisive role in shaping how individuals value regulatory goals within Gross's (2015) Extended Process Model (EPM), fundamentally influencing the trajectory of emotion regulation form{{sp}} identification through implementation. Because the EPM conceptualises regulation as a higher-order valuation process; one that evaluates an unfolding emotional response against contextually defined goals {{g}}. Cultural norms become central in determining which goals are activated and how strongly they are prioristised. Gross (2015) notes that valuation systems are &quot;highly influenced by prior experience and contextual variables,&quot; and cultural norms constitute one of the most pervasive of these variables. Briley's (2014) dynamic constructivist perspective further clarifies that [[W:Traditional knowledge|cultural knowledge]] is not monolithic but situationally activated: Individuals draw upon different cultural scripts depending on contextual cues, meaning that the valuation of regulatory goals within the EPM shifts fluidly across situations. This dynamic activation helps explain why the same strategy, such as suppression or reappraisal, can be valued positively in one cultural context and negatively in another.

{{RoundBoxTop|theme=6}}

''' Case Study - Cultural norms guiding emotional regulation in a workplace'''

Aiko, a 28‑year‑old engineer from Japan, is participating in a team meeting at her workplace in Tokyo. During the discussion, her supervisor critiques her recent project proposal, pointing out several weaknesses in front of the team. Aiko feels a sharp rise of embarrassment and disappointment, noticing her stomach tighten and her face flush. Despite the intensity of her emotional response, she keeps her expression neutral and responds softly: “Thank you for the feedback. I will revise the proposal.” She chooses to suppress her emotional reaction, maintaining composure to preserve group harmony and avoid disrupting the social cohesion of the team.

Meanwhile, her colleague Daniel, an American consultant visiting from the company’s New York office, receives similar critical feedback later in the meeting. He feels a comparable surge of frustration but responds differently. Daniel openly acknowledges his disappointment, explains his reasoning, and engages in a direct discussion about how the proposal could be improved. After the meeting, he reflects that “being upfront about how I felt made the conversation more productive,” demonstrating a cultural preference for authenticity and assertive emotional expression.
{{RoundBoxBottom}}

The outcomes of Chen et al.'s (2025) systematic review and meta-analysis {{ic|of how many studies?}} provide particularly compelling evidence for how cultural norms shape the valuation and effectiveness of regulatory goals. Their findings demonstrate that suppression is more maladaptive in &quot;indulgent and competition-driven cultures,&quot; where emotional expression is tied to autonomy, authenticity, and personal agency. In these contexts, the higher-level valuation system is more likely to appraise suppression as misaligned with culturally valued goals, increasing the likelihood of negative psychological outcomes. Conversely, Chen et al. (2025) show that suppression is less harmful, and sometimes adaptive in interdependent cultural contexts where emotional restraint supports relational harmony. This pattern illustrates how cultural norms calibrate the perceived utility of regulatory strategies: When harmony is culturally valued, suppression aligns with regulatory goals and is therefore more readily selected and more effective. Their analysis also reveals that reappraisal tends to be more beneficial in cultures characterised by high uncertainty tolerance and short-term orientation, suggesting that cultural norms shape not only the desirability of particular strategies but also the cognitive framing through which emotional situations are interpreted. These findings map directly onto the EPM's valuation architecture: cultural norms influence the identification of regulatory need, the selection of a strategy that aligns with culturally valued goals, and the implementation of that strategy that aligns with culturally valued goals, and the implementation of that strategy in ways that are socially coherent. Cultural norms sculpt the goals that guide higher-order valuation systems. {{g}} determining when regulation is necessary, which strategies are valued, and how regulatory success is defined. Ultimate {{g}}, cultural norms do not merely influence strategy choice, they shape the very architecture of emotional regulation, determining the meaning, purpose, and consequences of regulatory action. {{f}}

&lt;quiz display=&quot;simple&quot;&gt;
{Is Australia's culture considered interdependent?{{g}}
|type=&quot;()&quot;}
- True
+ False

{Has emotional suppression shown to be maladaptive in Australian culture?{{g}}
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;

== Cognitive Flexibility in Emotional Regulation ==
Cognitive flexibility plays a central role in emotion regulation because it enables individuals to adjust their regulatory efforts in response to shifting situational demands ((ic|Explain: is this part of the EPM?}}. Bonanno and Burton (2013) emphasise that regulatory flexibility rests on three core abilities: [[w:Context-sensitive|context sensitivity]] a diverse regulatory repertoire, and responsiveness to feedback. They define context sensitivity as “the ability to perceive impinging demands and opportunities from the situational context” and repertoire as “the ability to utilize a wide range of regulatory strategies that might accommodate divergent contextual demands and opportunities”. Cognitive flexibility supports these processes by allowing individuals to switch strategies when a chosen approach proves ineffective, thereby avoiding the “fallacy of uniform efficacy” in which any single strategy is assumed to be universally adaptive {{g}}. Recent neurophysiological evidence further demonstrates that switching decisions are guided by cognitive‑affective monitoring processes. Adamczyk et al. (2024) show that physiological markers such as corrugator EMG, the late positive potential (LPP), and stimulus‑preceding negativity (SPN) {{ic|only introduce abbreviations which are subsequently used}} predict when individuals abandon reappraisal in favour of distraction, indicating that flexible regulation is informed by real‑time feedback about regulatory success. As a result, individuals with greater cognitive flexibility tend to exhibit more resilient emotional trajectories, because the ability to update and adjust regulatory strategies promotes stable functioning across diverse and unpredictable contexts {{f}}. Together, these findings illustrate that flexibility, not the mere use of specific strategies, explains meaningful individual differences in regulatory effectiveness, with more flexible individuals showing better adjustment across varied emotional environments.

== Children and Emotional Regulation ==
[[File:ChildAngry.jpg|thumb|204x204px|'''Figure 2.''' Child expressing anger]]

Children demonstrate meaningful regulatory flexibility from an early age, showing both spontaneous strategy use and the ability to switch strategies when their initial approach proves ineffective (Parsafar et al., 2019). Parsafar et al. (2019) report that even young children spontaneously engaged in distraction and reappraisal, indicating that the foundations of flexible regulation emerge well before adolescence. As children age, they increasingly adopt cognitively demanding strategies such as reappraisal, reflecting developmental gains in working memory, attentional control, and abstract reasoning; (Parsafar et al., 2019). Negative emotion intensity also shapes children’s regulatory choices, with stronger emotional arousal prompting greater reliance on disengagement strategies like distraction, a pattern consistent with the model’s emphasis on context‑sensitive strategy selection (Parsafar et al., 2019). These developmental trends illustrate how the identification and selection stages of regulation become more refined across childhood, as children learn to evaluate emotional cues, choose strategies appropriate to situational demands, and adjust their responses dynamically. Understanding these normative patterns provides essential context for interpreting dysregulation in clinical groups

=== Children with autism spectrum disorder: ===
The Extended Process Model provides a useful framework for understanding why autistic youth are particularly vulnerable to emotion dysregulation and anxiety. Because the EPM conceptualises regulation as a multi‑stage system involving identification, selection, and implementation, disruptions at any of these stages can produce cascading regulatory difficulties. Conner et al. (2020) show {{how}} that autistic youth often struggle with accurately identifying emotional states and contextual cues, noting that “emotion dysregulation significantly predicted anxiety even after accounting for core autism symptoms,” suggesting that difficulties in the identification stage exert an independent influence on anxiety risk. Selection processes may also be compromised, as autistic youth frequently rely on narrow or rigid regulatory repertoires, limiting their ability to choose strategies that match situational demands. Finally, implementation can be hindered by cognitive load, sensory overwhelm, or reduced flexibility, all of which interfere with the dynamic adjustments central to the EPM’s higher‑order valuation system {{f}}. Taken together, these findings illustrate how the EPM helps clarify the mechanisms through which dysregulation emerges in autism: when the evaluative system responsible for monitoring and modifying emotional responses is disrupted, regulatory failures accumulate, producing the heightened anxiety profiles commonly observed in autistic youth.

== What Emotional Regulation Strategies are Most Effective {{g}} ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Types of strategies: ===
Emotion regulation encompasses several distinct families of strategies that differ in their mechanisms and cognitive demands:
* '''Situation‑focused strategies''' involve shaping one’s environment to influence emotional outcomes, either by choosing situations likely to elicit desirable emotions (''situation selection'') or by altering aspects of a situation to change its emotional impact (''situation modification'') (Gross, 2015).
* '''Attentional strategies''' regulate emotion by directing attention toward or away from emotional stimuli; distraction shifts attention away and reliably reduces negative affect, whereas sustained attention to emotional content—such as rumination—typically intensifies it (Strauss et al., 2016; Webb et al., 2012).
* '''Cognitive strategies''' operate by changing how emotional stimuli are interpreted, with approaches such as reappraisal, problem‑focused reframing, and perspective‑taking modifying the meaning of events and consistently supporting more adaptive emotional outcomes across cultures (Chen et al., 2025; Webb et al., 2012).
* '''Response‑modulation strategies''' target expressive, behavioural, or physiological responses after emotion has already unfolded, with suppression reducing outward expression but failing to diminish internal emotional experience and often increasing physiological activation (Webb et al., 2012; Chen et al., 2025). 

'''Table 2'''.

Types of regulation strategies {{ic|Use APA style captions i.e., first letter capitalisation}} {{f}}

{| class=&quot;wikitable&quot;
|'''Situation‑Focused Strategies'''
|Strategies that involve shaping or selecting environments to influence emotional outcomes (e.g., situation selection, situation modification).
|-
|'''Attentional Strategies'''
|Regulate emotion by shifting or stabilising attentional focus. Includes distraction (shifting attention away) and rumination (sustained attention on emotional content).
|-
|'''Cognitive Strategies'''
|Modify the meaning of emotional stimuli through reinterpretation. Includes reappraisal, problem‑focused reframing, and perspective‑taking.
|-
|'''Response‑Modulation Strategies'''
|Target expressive, behavioural, or physiological responses after emotion has unfolded. Includes expressive suppression and physiological modulation.
|}

=== Strategy effectiveness: ===
A wide range of emotion regulation strategies are available to individuals, each differing in cognitive demands, mechanisms, and contextual suitability. Situation‑focused strategies such as situation selection and situation modification allow individuals to shape their environments to influence emotional outcomes (Gross, 2015), while attentional strategies like distraction and mindfulness‑based attention regulate emotion by shifting or stabilising attentional focus. Cognitive strategies, including reappraisal, problem‑focused reframing, and perspective‑taking, modify the meaning of emotional stimuli and are consistently associated with better mental health outcomes across cultures (Chen et al., 2025). Further meta‑analytic evidence reinforces these distinctions: Webb et al. (2012) found {{how}} that cognitive change strategies, particularly reappraisal, reliably reduce emotional responding, whereas attentional strategies show more variable effects, with distraction producing reductions in negative affect but rumination exacerbating it. Response‑modulation strategies such as suppression target expressive or physiological responses after emotion has already unfolded, though suppression is widely linked to poorer wellbeing (Chen et al., 2025) and, as Webb et al. (2012) demonstrate {{how}}, tends to decrease outward expression without reducing internal emotional experience. Experimental evidence further clarifies how these strategies operate. Strauss et al. (2016) show {{how}} that distraction and reappraisal both reduce negative affect but rely on distinct mechanisms, with distraction engaging attentional disengagement and reappraisal requiring cognitively effortful reinterpretation. Neurophysiological findings highlight the importance of matching strategy demands to situational intensity: Adamczyk et al. (2024) demonstrate that switching to distraction enhances neural downregulation, indexed by reduced corrugator EMG and attenuated LPP, whereas switching to reappraisal can impair downregulation under high‑intensity emotional conditions due to increased cognitive load. Taken together, these findings show that no single strategy is universally optimal; instead, regulatory effectiveness depends on the alignment between contextual demands, cognitive resources, and the mechanisms through which each strategy operates. Distraction tends to be advantageous in high‑intensity contexts, while reappraisal supports longer‑term wellbeing when cognitive capacity and situational fit allow {{f}}.

==Conclusion==
Emotion regulation is best understood as a dynamic, context‑sensitive system, and the evidence reviewed in this chapter demonstrates that its effectiveness depends not on the use of any single strategy, but on the flexible coordination of identification, selection, and implementation processes. Across studies, reappraisal generally supports long‑term wellbeing, while suppression is consistently associated with poorer outcomes, yet these patterns are moderated by cultural norms, cognitive resources, and situational intensity (Chen et al., 2025). Experimental and neurophysiological findings further show that strategies such as distraction and reappraisal operate through distinct mechanisms, and that switching between them can either facilitate or impair downregulation depending on emotional load (Strauss et al., 2016; Adamczyk et al., 2024). Developmental research highlights that children begin to demonstrate regulatory flexibility early in life, with maturation of strategy selection and cognitive control shaping the emergence of more sophisticated regulation across childhood (Wong et al., 2019). These normative trajectories provide essential context for understanding dysregulation in clinical groups, particularly autistic youth, for whom difficulties in identifying, selecting, and adjusting regulatory strategies strongly predict anxiety beyond core diagnostic features (Conner et al., 2020). Taken together, the literature underscores that adaptive emotion regulation is not defined by the strategy itself but by the capacity to deploy strategies that fit the demands of the moment. This perspective not only clarifies individual differences in wellbeing but also offers a foundation for interventions aimed at strengthening regulatory flexibility across development and clinical contexts.

{{tip|Take home messages:
* Effective emotion regulation is flexible, not fixed; adaptability is more important than any single strategy.
* Cultural norms determine which regulatory goals are valued and therefore guide strategy selection within the EPM. They shape whether strategies like suppression or reappraisal are appraised as socially coherent or maladaptive.
* Cognitive flexibility enables individuals to shift goals and strategies as contexts change, making regulation more adaptive. Those with greater flexibility show more effective regulation because they can better align strategy choice with situational demands.
* Children have less developed identification, selection, and implementation processes, relying more on simple strategies. Adults show greater regulatory flexibility due to more mature cognitive control and metacognitive insight.
* Autistic youth often experience disruptions across all EPM stages, making it harder to identify emotions, select appropriate strategies, and adjust regulation. These difficulties allow emotional arousal to escalate, explaining the strong predictive link with anxiety.
*No single strategy is universally optimal; effectiveness depends on context, goals, and cognitive resources. The most adaptive regulation is flexible regulation, choosing strategies that fit the demands of the moment.
}}

== See also ==
* [[Motivation and emotion/Book/2024/ADHD and emotional regulation|ADHD and emotional regulation]] (Book chapter, 2024)
* [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|Cognitive strategies and emotion regulation]] (Book chapter, 2025)
* [[w:Emotional self-regulation| Emotional self-regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Emotion dysregulation |Emotional dysregulation]] (Book chapter, 2026)

==References==
{{Hanging indent|1=

Adamczyk, A. K., Koch, J., Miroslaw Wyczesany, Roelofs, K., &amp; van Peer, J. M. (2024). Emotion regulation flexibility: EEG/EMG predictors and consequences of switching between reappraisal and distraction strategies. Psychophysiology, 61(11). https://doi.org/10.1111/psyp.14646

Bonanno, G. A., &amp; Burton, C. L. (2013). Regulatory flexibility: An individual differences perspective on coping and emotion regulation. Perspectives on Psychological Science, 8(6), 591–612. https://doi.org/10.1177/1745691613504116

Briley, D., Wyer, R. S., &amp; Li, E. (2014). A dynamic view of cultural influence: A review. Journal of Consumer Psychology, 24(4), 557–571. https://doi.org/10.1016/j.jcps.2014.02.003

Chen, M. S., Cai, Q., Omari, D., Sanghvi, D. E., Lyu, S., &amp; Bonanno, G. A. (2025). Emotion regulation and mental health across cultures: A systematic review and meta-analysis. Nature Human Behaviour, 9. https://doi.org/10.1038/s41562-025-02168-8

Conner, C. M., White, S. W., Scahill, L., &amp; Mazefsky, C. A. (2020). The role of emotion regulation and core autism symptoms in the experience of anxiety in autism. Autism, 24(4), 931–940. https://doi.org/10.1177/1362361320904217

Gross, J. J. (2015). The extended process model of emotion regulation: Elaborations, applications, and future directions. Psychological Inquiry, 26(1), 130–137. https://doi.org/10.1080/1047840x.2015.989751

Parsafar, P., Fontanilla, F. L., &amp; Davis, E. L. (2019). Emotion regulation strategy flexibility in childhood: When do children switch between different strategies? Journal of Experimental Child Psychology, 183, 1–18. https://doi.org/10.1016/j.jecp.2019.01.004

Strauss, G. P., Ossenfort, K. L., &amp; Whearty, K. M. (2016). Reappraisal and distraction emotion regulation strategies are associated with distinct patterns of visual attention and differing levels of cognitive demand. PLOS ONE, 11(11), e0162290. https://doi.org/10.1371/journal.pone.0162290

Webb, T. L., Miles, E., &amp; Sheeran, P. (2012). Dealing with feeling: A meta-analysis of the effectiveness of strategies derived from the process model of emotion regulation. Psychological Bulletin, 138(4), 775–808. https://doi.org/10.1037/a0027600

}}

== External links ==
* [https://www.youtube.com/watch?v=gAMbkJk6gnE What are emotions? and why do we feel them?] (youtube)
* [https://www.youtube.com/watch?v=Yd6hR1qCfSM Emotion Regulation with James J. Gross, PhD] (youtube)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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    <title>Talk:Motivation and emotion/Book/2026/Extended process model of emotion regulation</title>
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      <comment>Book chapter feedback</comment>
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      <text bytes="21128" sha1="ndihypwiprw2xwxrc7rtw5gmkaemf3y" xml:space="preserve">== Gen AI statement ==
'''I have used''' the following Generative Artificial Intelligence (GenAI) tools/services in the

preparation of this assessment:

• GenAI service name and version: Microsoft Copilot version 153.0.4234.42 (Official build) (64-bit), Studiosity     

• Ways this GenAI service was used: GenAI was utilised to help format the essay and aid in the creation of key points. It was asked to then create separate paragraphs based on this, this was proofread, commented on, and refed into AI with fixes prepared. This again was personally vetted and rewritten, commented on and personally changed. This re-written version was then fed into Studiosity to check for grammar and spelling errors and feedback on the essay. The changes presented by Studiosity were personally changed, creating the finished product. 

I confirm that I have not used GenAI in the preparation of this assessment for any purpose other than what I have acknowledged above and I have cited and referenced any GenAI content in my assessment submission, applying the relevant referencing style. I understand that providing false or misleading information in this GenAI Acknowledgement Statement may constitute a breach of the University of Canberra (Student Conduct) Rules 2023.

== Recommendation for 'see also' section ==

Hi there, 

Great job on your topic development! I am writing a related chapter on [[Motivation and emotion/Book/2026/Emotion dysregulation|emotion dysregulation]], so I enjoyed reading your work thus far. 

I thought that the links currently listed in your ''See also'' section seem better suited to the ''External links'' section, [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2832166&amp;oldid=2830650 which I have added for you]. It may be worth relocating those links, and including some relevant book chapters or Wikipedia pages instead (e.g., [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|cognitive strategies and emotion regulation]] (Book chapter, 2025)). 

Best regards.  [[User:U3285438|U3285438]] ([[User talk:U3285438|discuss]] • [[Special:Contributions/U3285438|contribs]]) 04:33, 8 September 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:TheHutt02|Hi [[User:TheHutt02|TheHutt02]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:24, 11 September 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Consider, for example, adding sub-headings to the key section which is currently titled: &quot;The extended process model of emotional regulation&quot;
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Key points are well developed for each section
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- Theory and research --&gt;
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
&lt;!-- Other --&gt;
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Consider using Studiosity Writing Feedback+ or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft.
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented
&lt;!-- Caption --&gt;
# Figure caption(s) should explain how the image connects to key points being made in the main text
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice.
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place each quiz question in the most relevant section
# Focus the quiz question(s) on the take-home messages
&lt;!-- Tables --&gt;
# Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Good
&lt;!-- Systematic reviews --&gt;
# At least one relevant systematic review and/or meta-analysis has been identified
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## [[Help:Wikitext quick reference|italicisation]]
## include hyperlinked dois
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- External links --&gt;
# External links
## Basic
## Only include links directly related to the sub-title
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Include source in brackets after link
## Use alphabetical order
|9=
&lt;!-- User page --&gt;
# Simple but effective
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:24, 11 September 2026 (UTC)

== Recommendation: The Extended Process Model of Emotional Regulation ==
Hello, 

Great job so far. I think your overview was very thorough and engaging, though perhaps a bit long so you may want to condence it. 

My biggest concern in regard to the book chapter is your 'extended process model of emotional regulation' section. It's dense and covers a lot of content, which is good, but can be unengaging if it's just big blocks of texts. I'd consider breaking up the content with highlight boxes and key points/features that summarise the major components of this model whilst still being visually engaging for the reader. I'd also consider adding a diagram or some sort of visually representative figure so a person can mindmap how the model works. 

Great job! [[User:Monuc9|Monuc9]] ([[User talk:Monuc9|discuss]] • [[Special:Contributions/Monuc9|contribs]]) 01:57, 9 October 2026 (UTC)

&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is a basic chapter
# The main area(s) for potential improvement:
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested) 
&lt;!-- Overall - Edit summaries --&gt;
#* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
&lt;!-- Overall - GenAI --&gt;
#* GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
&lt;!-- Overall – Citations --&gt;
# Promising use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2844762&amp;oldid=2839787 these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Basic scenario or case study
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Make it more clear which aspect(s) of the EPM the scenario illustrates
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
&lt;!-- Overview – Explains problem --&gt;
# Solid, if somewhat abstract, description of the psychological problem or phenomenon
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are problematic
# Add a focus question that allows the EPM to be explained in clear, basic terms
# Most of the focus questions are about applied aspects of the topic that go beyond the topic in the sub-title; they are not irrelevant, but they are mostly not core (i.e., they are tangential) to the topic
# Use [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
&lt;!-- Theory – Builds on --&gt;
# Builds reasonably well on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Reasonably good depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Good use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Basic use of examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Basic review of relevant research
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# More detail about key studies would be ideal
# In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
#  of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Basic integration between the most relevant theory(ies) and the best research
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Very good/ summary and conclusion
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
&lt;!-- Style – Layout --&gt;
# Layout
## The headings could be more clearly aligned with the focus questions
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## See earlier comment about [[#Heading casing|heading casing]]
## Remove colons (:) from the ends of headings
## Include an introductory paragraph before branching into sub-sections
&lt;!-- Style – Grammar --&gt;
# Grammar
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Brief captions; provide more detail to help connect the figure to the text
&lt;!-- Style – Figures - Citations --&gt;
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
&lt;!-- Style – References --&gt;
## The list of references is surprisingly short
## References use good APA style:
### Check and correct use of italicisation
### Page numbers are missing for some references
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Basic use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Basic use of figure(s)
# Reasonably good use of table(s)
# No use of feature boxes other than the opening scenario
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Basic use of quiz(zes) and/or reflection question(s)
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
&lt;!-- Learning features – See also --&gt;
# Very good use of the [[{{PAGENAME}}#See also|See also]] section
## Add more links
&lt;!-- Learning features – External links --&gt;
# Insufficient use of the [[{{PAGENAME}}#External links|External links]] section
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Add more links
|9=
&lt;!-- Social contribution comments... --&gt;
# ~1 logged minor contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~2 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 6 November 2026 (UTC)</text>
      <sha1>ndihypwiprw2xwxrc7rtw5gmkaemf3y</sha1>
    </revision>
    <revision>
      <id>2844767</id>
      <parentid>2844763</parentid>
      <timestamp>2026-11-06T22:10:36Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2844767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20902" sha1="4of7yi9d6jnmbo9t5fep5tqh8suz7fg" xml:space="preserve">== Gen AI statement ==
'''I have used''' the following Generative Artificial Intelligence (GenAI) tools/services in the

preparation of this assessment:

• GenAI service name and version: Microsoft Copilot version 153.0.4234.42 (Official build) (64-bit), Studiosity     

• Ways this GenAI service was used: GenAI was utilised to help format the essay and aid in the creation of key points. It was asked to then create separate paragraphs based on this, this was proofread, commented on, and refed into AI with fixes prepared. This again was personally vetted and rewritten, commented on and personally changed. This re-written version was then fed into Studiosity to check for grammar and spelling errors and feedback on the essay. The changes presented by Studiosity were personally changed, creating the finished product. 

I confirm that I have not used GenAI in the preparation of this assessment for any purpose other than what I have acknowledged above and I have cited and referenced any GenAI content in my assessment submission, applying the relevant referencing style. I understand that providing false or misleading information in this GenAI Acknowledgement Statement may constitute a breach of the University of Canberra (Student Conduct) Rules 2023.

== Recommendation for 'see also' section ==

Hi there, 

Great job on your topic development! I am writing a related chapter on [[Motivation and emotion/Book/2026/Emotion dysregulation|emotion dysregulation]], so I enjoyed reading your work thus far. 

I thought that the links currently listed in your ''See also'' section seem better suited to the ''External links'' section, [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2832166&amp;oldid=2830650 which I have added for you]. It may be worth relocating those links, and including some relevant book chapters or Wikipedia pages instead (e.g., [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|cognitive strategies and emotion regulation]] (Book chapter, 2025)). 

Best regards.  [[User:U3285438|U3285438]] ([[User talk:U3285438|discuss]] • [[Special:Contributions/U3285438|contribs]]) 04:33, 8 September 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:TheHutt02|Hi [[User:TheHutt02|TheHutt02]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:24, 11 September 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Consider, for example, adding sub-headings to the key section which is currently titled: &quot;The extended process model of emotional regulation&quot;
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Key points are well developed for each section
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- Theory and research --&gt;
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
&lt;!-- Other --&gt;
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Consider using Studiosity Writing Feedback+ or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft.
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented
&lt;!-- Caption --&gt;
# Figure caption(s) should explain how the image connects to key points being made in the main text
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice.
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place each quiz question in the most relevant section
# Focus the quiz question(s) on the take-home messages
&lt;!-- Tables --&gt;
# Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Good
&lt;!-- Systematic reviews --&gt;
# At least one relevant systematic review and/or meta-analysis has been identified
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## [[Help:Wikitext quick reference|italicisation]]
## include hyperlinked dois
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- External links --&gt;
# External links
## Basic
## Only include links directly related to the sub-title
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Include source in brackets after link
## Use alphabetical order
|9=
&lt;!-- User page --&gt;
# Simple but effective
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:24, 11 September 2026 (UTC)

== Recommendation: The Extended Process Model of Emotional Regulation ==
Hello, 

Great job so far. I think your overview was very thorough and engaging, though perhaps a bit long so you may want to condence it. 

My biggest concern in regard to the book chapter is your 'extended process model of emotional regulation' section. It's dense and covers a lot of content, which is good, but can be unengaging if it's just big blocks of texts. I'd consider breaking up the content with highlight boxes and key points/features that summarise the major components of this model whilst still being visually engaging for the reader. I'd also consider adding a diagram or some sort of visually representative figure so a person can mindmap how the model works. 

Great job! [[User:Monuc9|Monuc9]] ([[User talk:Monuc9|discuss]] • [[Special:Contributions/Monuc9|contribs]]) 01:57, 9 October 2026 (UTC)

&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is a basic chapter
# The main area(s) for potential improvement:
#* provide more detailed review of the best psychological research about the topic
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
&lt;!-- Overall - Edit summaries --&gt;
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
&lt;!-- Overall - GenAI --&gt;
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
&lt;!-- Overall – Citations --&gt;
# Promising use of academic, peer-reviewed citations to support claims
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2844762&amp;oldid=2839787 these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Good
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Basic scenario or case study
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# Make it more clear which aspect(s) of the EPM the scenario illustrates
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
&lt;!-- Overview – Explains problem --&gt;
# Solid, if somewhat abstract, description of the psychological problem or phenomenon
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are problematic
# Add an initial focus question that allows the EPM to be explained in clear, basic terms
# Some of the focus questions are about applied aspects of the topic that go beyond the topic in the sub-title; they are not irrelevant, but they are applied aspect of, rather than core to, the topic
# Use [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Reasonably good—relevant theory is selected, described, and applied, with some room for improvement
&lt;!-- Theory – Builds on --&gt;
# Builds reasonably well on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Reasonably good depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Good use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Note that if you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Basic use of examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Reasonably good review of relevant research
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# More detail about key studies would be ideal
# In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
#  of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Reasonably good integration between the most relevant theory(ies) and the best research
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent summary and conclusion
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is reasonably good, but appears to be based on overuse of genAI content and/or direct quotations; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia] (relates to the quiz questions).
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
&lt;!-- Style – Layout --&gt;
# Layout
## The headings could be more clearly aligned with the focus questions
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## See earlier comment about [[#Heading casing|heading casing]]
## Remove colons (:) from the ends of headings
## Include an introductory paragraph before branching into sub-sections
&lt;!-- Style – Grammar --&gt;
# Grammar
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Brief captions; provide more detail to help connect the figure to the text
&lt;!-- Style – Figures - Citations --&gt;
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
&lt;!-- Style – References --&gt;
## The list of references is surprisingly short
## References use good APA style:
### Check and correct use of italicisation
### Page numbers are missing for some references
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Basic use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Basic use of figure(s)
# Reasonably good use of table(s)
# No use of feature boxes other than the opening scenario
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Basic use of quiz(zes) and/or reflection question(s)
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
&lt;!-- Learning features – See also --&gt;
# Very good use of the [[{{PAGENAME}}#See also|See also]] section
## Add more links
&lt;!-- Learning features – External links --&gt;
# Insufficient use of the [[{{PAGENAME}}#External links|External links]] section
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Add more links
|9=
&lt;!-- Social contribution comments... --&gt;
# ~1 logged minor contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~2 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 6 November 2026 (UTC)</text>
      <sha1>4of7yi9d6jnmbo9t5fep5tqh8suz7fg</sha1>
    </revision>
  </page>
  <page>
    <title>Social Victorians/Boxing/Smith-Slavin 1889-12-23</title>
    <ns>0</ns>
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    <revision>
      <id>2844794</id>
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      <timestamp>2026-11-07T00:02:51Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <origin>2844794</origin>
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      <text bytes="184214" sha1="0i88lw4v45m6h1fceettjqqrq12g9w4" xml:space="preserve">=Event=
International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made to offer complete coverage.

==Logistics==
* 23 December 1889, 8:00 a.m.
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
# Jem Smith, British boxer
# Frank Slavin, Australian boxer
# Lord Rosslyn
# Sir M. Sandys
# Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology)
==Entertainment==

==Anthology==
The Sportsman

The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 (7 columns) entirely to this story.


This reporter's writing style is very ornate, and he also uses slang and local vocabulary: sabots, Brums, 

Newspaper Title: The Sportsman 24 December 1889, Tuesday: 3 [of 4], Col. 1a–7c [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.

THE GREAT FIGHT.

SMITH &amp; SLAVIN FOR £1000.

DISORDERLY PROCEEDINGS.

BRUTAL ATTACK ON SLAVIN.

SMITH KNOCKED DOWN REPEATEDLY.

HE LEAVES THE RING.

A DRAW DECLARED AFTER 14 ROUNDS.

The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.

THE JOURNEY TO THE CONTINENT.

(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few  qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of THE SPORTSMAN I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a surplusage of note-books dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey, who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity.

Pugilism at the Piano.

While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys coon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arnuigements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out agent. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. 

THE FIGHT. 

First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavic walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking beck to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, lmin 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite comer, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion wars something terrible, and the ring looked very much like being invaded. The ropes in Smith's comer were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth.  Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact  should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case  of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old  style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a]  

Page Number: 4

that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow.  Bearing on this subject, Rule 26 says: — &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side.

SLAVIN'S OPINION OF HIS TREATENT. 

His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. 

A Testimonial for Slavin. 

So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sia, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. 

Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation] 

Eyewitnesses Interviewed. 

A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which  did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds.  Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst  characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, isasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his  party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought far two hours more. He could have beaten Smith and both his seconds into the  bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He bad to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot; 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen  have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; 

Reception of the News in Birmingham. 

Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a]  and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell  was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, wa unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', lor the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names.   

Frank Hinde (for Frank Slavin).   

John Fleming (for James Smith).   

Witness for Smith, F. Montague.   

Witness for Stavin, John Lewes.   

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.   

The Men in Training   

Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson.   

Finding the Battle-Ground.   

(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot;  he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. '''let the pony have her head''', and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do yen earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pictch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges.

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.

Performance of the Men.  Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.'' 

FRANK P. SLAVIN. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight '''Laing, the Champion'''. Tams all net be weed wee until  they bed ° MOW dads sheet In early weds. The  C ` were then seedited ler Vir aide. sad lathe with  Ilse gnaws at Weeiplal. Wag bliss le Ness op el the  sag at the NW mewl. Alfas 0. we went but to  114MMNIVOI4 ea Vz le AM ea a swish with Jai..  eat um Is is he MI peas le Amorist.  Si ma Ifrafes. but Mhz EWE bed  P hid lb s w il a h MA latheowalest woodoeirarleibew. Nibs  vas wow in !AM. MI bowel saltallimps te  This be midi kr tO will small&amp;es kse whiii  Sd le linhe i a md impolarl oat ha Imia  way bed, and all hy eras to a aunty  half en lam masers be Wasener, al Das  eEwi pet mond liallmoos la Me piper. em Tidier  le V a le as numb; INC sae drla dy  aria il= 4iis Ldemelset Oberla seme arem s s  Er Lista ihrialser air 7 lire ;was lisesilumasio.  lashes Me bused • half. awairially We lee Nkomosim  Mao wee mime. drawee all bobs se, sal Os Ws  =with lea gimes. sod mosed tb. sea.  min wiN mama to the mimic to the ill sena  Pram to this ally_Tame, the tesqemeret  Hula, fee NM., with the result that the Mee  addedleast to hie ale. Tana yam  tat mower*  the w =tisee la the third read. TIP, to  ealdwal to be aet Met perternann. Atom  this tins Wien lleadoE. fie IlliNsobapel. woo le NM  beam, mid fashadyholy of bashms. they pet op • hat  or lOU he a Leaser le ben ler lid. sades  After wafters blue pa* eV Portet however. herder.  Mauls were Pp a dear with the watch. sal Sin  the &quot;army.&quot; a ease *say  ever ts, Rill hr . we star followed, end then Star we  Ora IT a hie Yet appear* la &amp;whoa at  Adry's To where he beat &quot; Oberwaid &quot; Goode to  la rear le • Era of twelve reads. for 400/. •  THE PELICAN CLUB AND THE FIGHT.  TO THE EDITOR OP THE SPORTSMAN.  SIR,—As an impreasion prevails that the Smith and  Slann fight wan arranged under the patronage of this  club, 1 should esteem it a favour if you would allow me  to state emphatically that the Pelican Club gave no  countenance to the affair in any way whatever, and  had nothing to do with it.—Yours truly,  CECIL RALEIGH, Hoe. Sec.  Pelican Club, 21, Denman-street, W., Dec. 23.  We hear on good authority that it is the intention  el the mambas of the Pelican Club to present Slavin,  the Australian Champion, with a belt, in recognition  of the manly exhibition he gave yesterday at Bruges.  OF TEE  FRANK SLAVIN TO JOHN L.  SULLIVAN.  FRANI StAvitt, the Champion of Australia, is prepared  to meet John L. Sullivan, under London Prize  Ring Rules, for 500/. or 10004 a-side and the  Championship of the World. Slavin bay lodged  with no a sum of 1001., and will cosiest  the title with Sullivan in either America or  Australia in six mouths from signing articles. We  have '''cabled''' this challenge to Mr R. K. Foe, of New  York, and the money lodged with him or the Editor  of Tsui Smarms's well lead to a match. 

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn, Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
{{reflist}}</text>
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  </page>
  <page>
    <title>User:~2026-56788-08/medical issues</title>
    <ns>2</ns>
    <id>332700</id>
    <revision>
      <id>2844734</id>
      <parentid>2844504</parentid>
      <timestamp>2026-11-06T19:28:16Z</timestamp>
      <contributor>
        <username>~2026-56788-08</username>
        <id>3113246</id>
      </contributor>
      <origin>2844734</origin>
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      <format>text/x-wiki</format>
      <text bytes="732" sha1="l735alnlcopq7nkr24b7vjgstjk7v6x" xml:space="preserve">== als ==

CBC the fifth estate Presents: ALS Country

One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease.

check out the following:
* [[RNA interference/Medical#section Treating Lou Gehrig's disease]]
* [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]]
* [[World War I -- Life Histories/Section 018/H. Gaston Carney]]

[[w:ALS|enwp article about Amyotrophic lateral sclerosis (ALS)]]

== Cancer and exercise ==
&quot;Continuing to exercise after a cancer diagnosis can significantly improve post-surgical recovery, reduce recurrence rates, and increase overall survival&quot;
NHK medical frontiers

== womens health ==</text>
      <sha1>l735alnlcopq7nkr24b7vjgstjk7v6x</sha1>
    </revision>
    <revision>
      <id>2844737</id>
      <parentid>2844734</parentid>
      <timestamp>2026-11-06T19:31:19Z</timestamp>
      <contributor>
        <username>~2026-56788-08</username>
        <id>3113246</id>
      </contributor>
      <comment>/* womens health */ Women’s Health Issues Doctors Too Often Miss</comment>
      <origin>2844737</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="836" sha1="q5pn35r9mj5zz41fx3krwy7wo4du3pi" xml:space="preserve">== als ==

CBC the fifth estate Presents: ALS Country

One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease.

check out the following:
* [[RNA interference/Medical#section Treating Lou Gehrig's disease]]
* [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]]
* [[World War I -- Life Histories/Section 018/H. Gaston Carney]]

[[w:ALS|enwp article about Amyotrophic lateral sclerosis (ALS)]]

== Cancer and exercise ==
&quot;Continuing to exercise after a cancer diagnosis can significantly improve post-surgical recovery, reduce recurrence rates, and increase overall survival&quot;
NHK medical frontiers

== womens health ==
Women’s Health Issues Doctors Too Often Miss with CNN’s Pamela Brown
Pamela Brown CNN YouTube video</text>
      <sha1>q5pn35r9mj5zz41fx3krwy7wo4du3pi</sha1>
    </revision>
    <revision>
      <id>2844740</id>
      <parentid>2844737</parentid>
      <timestamp>2026-11-06T19:42:15Z</timestamp>
      <contributor>
        <username>~2026-56788-08</username>
        <id>3113246</id>
      </contributor>
      <comment>/* womens health */ === estrogen patches ===</comment>
      <origin>2844740</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="981" sha1="s4gc25pxrek0372b9r1xo59saquwgcx" xml:space="preserve">== als ==

CBC the fifth estate Presents: ALS Country

One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease.

check out the following:
* [[RNA interference/Medical#section Treating Lou Gehrig's disease]]
* [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]]
* [[World War I -- Life Histories/Section 018/H. Gaston Carney]]

[[w:ALS|enwp article about Amyotrophic lateral sclerosis (ALS)]]

== Cancer and exercise ==
&quot;Continuing to exercise after a cancer diagnosis can significantly improve post-surgical recovery, reduce recurrence rates, and increase overall survival&quot;
NHK medical frontiers

== womens health ==
Women’s Health Issues Doctors Too Often Miss with CNN’s Pamela Brown
Pamela Brown CNN YouTube video

=== estrogen patches ===
www.cnn.com /2026/10/06/health/video/cnn-sitroom-pamela-brown-estrogren-patch-shortage-mary-claire-haver-womens-health</text>
      <sha1>s4gc25pxrek0372b9r1xo59saquwgcx</sha1>
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  <page>
    <title>Cosmology: UG.NUG an Observation-Based Model</title>
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      <id>2844720</id>
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      <timestamp>2026-11-06T13:34:52Z</timestamp>
      <contributor>
        <username>Wb525</username>
        <id>3112992</id>
      </contributor>
      <comment>Text edition chap UG.NUG redshift</comment>
      <origin>2844720</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35854" sha1="0esirf73j2tr6zqnimzyyjxkvndmxnd" xml:space="preserve">{{Research project}}
Original research project by Willy Bohane — UG.NUG.
https://doi.org/10.5281/zenodo.20644064

== All the texts in this presentation ==

[[Cosmology: UG.NUG an Observation-Based Model|UG.NUG]]

[[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]]

[[Cosmology: UG.NUG an Observation-Based Model/Cosmic Time|Cosmic Time]]

[[Cosmology: UG.NUG an Observation-Based Model/the Mass Imbroglio|the Mass Imbroglio]]

[[Cosmology: UG.NUG an Observation-Based Model/the CMB|of the CMB]]

[[Cosmology: UG.NUG an Observation-Based Model/the GRID|of the GRID]]

[[Cosmology: UG.NUG an Observation-Based Model/Gravity|of Gravity]]

[[Cosmology: UG.NUG an Observation-Based Model/Black Holes|of Black Holes]]

= Introduction =
This article presents a reading of the Universe that differs in every respect from the existing model. The latter contains a false start which, for reasons explained in the dedicated article on [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]], renders it null and void from the very first millisecond of its narrative: the Singularity.

Before reading this detailed text, whose links to technical articles support and further develop the arguments presented, readers can consult a simplified [https://www.wbohane.com/BLOG/!three_point_universe.aspx three-point diagram] offering an overview of the proposed simplicity of the Universe.

= State of Play =
Before anything else, it is important to understand that cosmology, in its interpretative function, is not an exact science. As its name indicates, it means, in Greek, ''the study of the world'' — and therefore, by extension, its origin, its structure, its evolution, its content, and its future. 

This is why it resembles more a religion than science. As with religion, everybody can say anything, everybody can believe anything, everybody can make anyone believe anything. Humanity is divided into many creeds, and so is cosmology. 

This is why the following list of flaws is possible, especially where science is subjected to the rules of a free market place.

'''[[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|The long long list of ΛCDM flaws]]'''

= Observation, Reading, Reflection =
In accordance with the principle of parsimony, these should be the first instincts to guide us when we look up at the sky and ask ourselves what the Universe is. But we must proceed with due respect for our own nature as human beings, confronted with the immensity of a habitat we did not choose, which brings us into existence, sustains us, and ultimately eliminates us without even noticing. In this sense, cosmology has less to do with technology than with philosophy, the mother of all sciences.

When human beings, through the [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]] model, presume to decide when the Universe began and how it began, they brazenly violate the modesty and humility that should be incumbent upon them. They do so much as the authors of the biblical creation narrative did—distant predecessors whose audacity might be forgiven, but whose example is no longer intellectually acceptable in the twenty-first century.

The issue here is not even to revisit the invalidity of such a conjecture, which is discussed at length in the list of flaws. Rather, it is to ask by what right the model's architects permitted themselves to make such an extraordinary claim in the first place. From that moment on, the claim becomes null and void, without further appeal.

= the Universe =
So, what does observation tell us when we look up at the sky?

First and foremost, it tells us that the Universe exists, that it was already here before us. Our age and our experience also tell us that, as human beings, we are merely passengers, temporary workers, and mortals in waiting, and that whatever we do, the Universe will always be stronger than we are.

Consequently, the only natural and parsimonious conclusion we can draw from looking at the sky is that the Universe exists, that it did not ask us to exist, and that it has never told us when it began, how it began, or why it began.

From this starting point, we can say, without doubt, without prejudice, without invention, and, quite simply, without making things up, that what we see around us is the Universe; that it is the only thing that exists as far as we are concerned; that it does not know human clock-measured time; and that it is absolutely impossible for us to say anything more about it.

Pure logic would therefore suggest that we consider ourselves inhabitants of a stationary Universe, one that would have absolutely no need to expand. The initial observation does not impose expansion upon us. A stationary universe therefore constitutes the most parsimonious starting point. On the one hand, this means not inventing an improbable birth date; on the other, it means not inventing an equally improbable expansive Big Bang.

However, with our eyes fixed on the sky, especially at night, we see the Moon. And we are entitled to wonder why it does not fall on our heads, even before knowing about gravity or having the slightest idea of its existence.

The natural reflex would therefore be to consider the concept of the Apeiron — an undefined, omnipresent substrate — or Aristotle's concept of Ether, or Faraday's idea of a &quot;something&quot; that might sustain celestial objects.

Although these concepts are not entirely identical, they all point towards the same fundamental intuition: the possibility of a &quot;Something&quot; underlying the visible world. This may well be one of humanity's oldest intellectual reflexes. It comes naturally to mind whenever something appears to hang in the void, even during a Houdini performance. Something must be holding it, surely!

And at this point, one might begin to think that the Universe is not what we see, but rather what we do not see: that what is visible emerges from a fertile, invisible substrate that fills the entire Universe. This would be a very concrete and believable image if all visible celestial objects appeared to be suspended in the void — suspended, yet held in the embrace of that same substrate, which prevents everything from collapsing. 

It is much like the air that fills the space on Earth. When the plant emerges from the ground, the air surrounds the plant, continues to envelop it, nourish it, and influence its nature. 

= The CMB =
If we remain within this parsimonious perspective, which so far has required no invention, no additions, and no patches, it is easy to see why a sense of serenity might take hold of our minds. We have already saved ourselves quite a lot: the singularity and its strangeness, the improbable Big Bang, expansion and the many complications it brings in its wake… But be patient. We are about to do even better.

Because when Penzias and Wilson discovered the cosmic microwave background radiation, nothing in that discovery compelled us to think of a fossil. The idea of a distant past was not even a necessary intellectual or cosmological option within this line of thought. Why go looking for a fossil, and where would it have come from? No, the question would not even arise. We would instead think of that substrate — the very one we had already invoked to explain why the Moon does not fall on our heads — which might now turn out to be noisy as well. 

Hardly a reason to spoil the party, this radiation would be the living confirmation of what humanity had intuited for 2,700 years: a &quot;Something&quot; that sustains &quot;Everything.&quot; So what better response than to shout: &quot;Bingo!&quot;

Since none of the observed properties of the [[Cosmology: UG.NUG an Observation-Based Model/the CMB|CMB]] contradicts this possibility, it would be only fair, and consistent with the principle of parsimony, to acknowledge that this interpretation is far simpler and more appealing than the intricate story of a cosmic fossil, its journey across billions of years, its puzzling temperature, and all the complications that come with it.

= the Arrow of Emergence =
A glass is the only thing the universe deigned to offer us before its long nap. Not the glass itself, but what it whispers when it breaks. When it slips from our hands and, by misfortune, the floor is tiled rather than carpeted, it shatters. And if we read the universe simply, we catch its message — the message of the irreversible Cosmic Order: the one that prevents the glass from reassembling, the one that prevents events from denying their own existence.

Thus, the universe gives us the direction from past to future, what current physics commonly calls the Arrow of Time. It is striking: it gives us the Arrow of Time, but not time itself.

The current name, Arrow of Time, is not really appropriate, since it is not “time” in the chronometric sense. It is rather, as explained earlier, a pre‑cosmic Order that governs everything, the universe included. Yes, this is an inescapable consequence of a careful reading of the universe. This succession order is of primary importance in the UG.NUG reading, but also in the ΛCDM, which — it seems — has not given it the place it deserves.

In the [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]], one observes that the universe ceased its creative activity only once its work was complete. One day, it emerged from an unknown state — the term ''Big Bang'', which everyone knows, usually refers to this event — and then particles appeared, followed by atoms, stars, galaxies… Since then, nothing is created, nothing is lost, everything transforms, as Lavoisier would say.

What matters here is not the validity of this narrative, but the Succession  of things: this natural succession of phases is precisely the Succession  in question. And that is why we have renamed this arrow the Arrow of Emergence. The misunderstanding dissolves, and the meaning becomes clearer: Succession  of emergences versus time‑keeping.

The Arrow of Emergence therefore indisputably precedes the universe. It’s new, fresh off the shelf, ready‑to‑wear.

This can mean several things:

* the arrow was there, and that’s all. We accept the universe and take the arrow as a bonus,
* the arrow was there, and therefore the universe is governed by something we do not know,
* the arrow was there, and therefore our universe is part of a series of universes,
* the arrow was there, and therefore we are free to consider any interpretation.

In short, it mainly means that the Arrow of Emergence is particularly important — especially for what follows.

= The GRID =
Since CMB is the name given to the discovery, it seems fair to rename it according to its cosmological function. Faraday deserves his “something,” and we have chosen [[Cosmology: UG.NUG an Observation-Based Model/the GRID|GRID]], which stands for Global Reference in Dynamics.

In UG.NUG, the model presented in this paper, the [[Cosmology: UG.NUG an Observation-Based Model/the GRID|GRID]] is ''le clou du spectacle'', as the French would say — the highlight of the entire show. It is the key element that dissolves most of the paradoxes of ΛCDM in such a simple way that it seems almost impossible to have overlooked it.

In other words, without the GRID, UG.NUG would be no more interesting than any other model.

= UG =
Apart from its resemblance to the exclamation of a prehistoric man, this name carries a very important cosmological meaning.

What we call geometry refers to the shape or size of an individual object, properties that matter in its interactions with reality. Geometry begins above the level of chemistry.

UG stands for the French '''''Uniformité Genrée''''' (Categorical Uniformity). We coined this term to describe a phenomenon whose significance has gone unnoticed.

In short, everything produced by the Universe in its earliest stages — particles, nuclei, and so on — belongs to this regime, in which objects possess no geometry. They have no individuality, no history, and no variation. This uniformity is the hidden foundation of everything: the silent order governing the birth of the Universe.

For now, let us simply understand that, in this model, the world of the very small — below the atomic level — has no geometry, even though some textbooks may suggest otherwise.

= NUG =
NUG stands for the French '''Non-Uniformité Genrée''' (Categorical Non-Uniformity). It represents everything that possesses geometry and individuality: stars, planets, meteors, and, of course, everything that derives from them — humans, artifacts, flora, and fauna.

No two pebbles are alike, nor are any two trees. Individuality increases with complexity.

The transition from UG to NUG occurs through the assembly of atoms and molecules, the underlying mechanism of which remains unknown.

NUG is the third and final phase in the emergence of an object.

Since humans, along with their clocks and measuring instruments, belong to NUG, the time measured by our wristwatches is also a NUG phenomenon. In this model, measurable time exists only where geometry and individuality exist.

= UG.NUG =
UG.NUG is the name of this model. It encompasses the two regimes that govern the entire Universe. 

As we shall see, NUG — and therefore geometry — emerges later than UG. Once it appears, however, it becomes an integral part of the Three Point Universe, a concise schematic representation of the UG.NUG model.

NUG encompasses all objects, including the Moon and the other celestial bodies mentioned earlier, which, fortunately, do not fall on our heads.

== Redshift ==
The UG.NUG reading understands that geometry is not a starting point, but a secondary level of organization, appearing only after UG. This single shift is enough to dissolve almost all cosmological problems of the ΛCDM.

Thus, the universe does not begin with a geometric explosion, but with an undifferentiated background. Geometry appears only when objects cease to be identical. This conceptual reversal eliminates in one stroke the paradoxes related to the origin of space, its shape, its homogeneity, or its flatness. One no longer tries to explain why geometry is already there. It is not there at the beginning. It appears naturally, carried by the Arrow of Emergence, and stabilizes thanks to the GRID.

The image of a fabric tube being inflated illustrates this perfectly. The tube already exists, but it is not yet deployed. Near the opening, the fabric is inflated; farther away, it seems to inflate progressively; in the distance, it is still flat. For an observer inside, this creates the illusion of expansion, then accelerated expansion, even though it is only the progressive realization of the structure. The distant regions appear to recede because they are not yet realized, not because they are moving. This inversion of perspective dissolves the very idea of cosmic expansion, inflation, an initial singularity, or dark energy.

The spectroscopic observations and measurements themselves are not being disputed. The observed redshifts, the recorded spectra, and the data themselves remain unchanged. What UG.NUG challenges is the interpretation placed upon them: that redshift necessarily indicates recession, and that this recession necessarily means that space is expanding. The issue, therefore, is not the validity of the observations, but the assumptions through which they are interpreted. The same measurements may point to a different reading of the Universe.

There is, however, a further question concerning the validity of the measurements themselves. As discussed in the paper on Cosmic Time, if the Universe has no intrinsic time, the very notion of measuring cosmic durations or assigning chronological ages to distant events becomes questionable. Such measurements rely on a chronometric framework whose applicability UG.NUG challenges. This raises a separate and much broader issue, which cannot be addressed here without taking us away from the subject of redshift.

= the Genesis Phases =
Now that the framework has been established, and we have seen that our universe is neither outlandish nor fundamentally different from the one we know, and that it is not threatened by our models, however incongruous they may be, we can begin to imagine what might actually have happened to produce the spectacle before our eyes. 

For this section, then, a little imagination is indeed required. Yet, for reasons of pure logic, the sequence you are about to discover is hardly surprising. The principle of parsimony remains our guide. Water has always flowed, and will always flow, along the paths best suited to its passage. This natural law, which is suspected to have inspired William of Ockham in formulating his principle, is far from trivial when attempting to reconstruct an event such as the emergence of the universe through a virtual model. We shall therefore continue to refrain from introducing artificial devices into the natural stages described below.

= the First Phase =
Strictly speaking, this is not a phase at all. It is included here to make clear that everything begins with the omnipresence of the universe. This phase never began, because it is our sole reality. Nothing else exists.

This omnipresence is inferred from, and embodied in, the omnipresent GRID, which belongs to the UG regime, where objects inaccessible to our senses have no history, variation, or individuality.

We do not know what it is made of. Nevertheless, it is logical to suppose that it contains the codes of the perceivable universe: the non-geometric component commonly encountered at the depths of matter and described by quantum mechanics, including elementary particles and atomic nuclei. For human beings, such a configuration borders on metaphysics. Our cognitive faculties are not designed to grasp invisible, non-geometric entities that carry information, almost like ghosts. We can invent ghosts, but we cannot manage them.

According to the standard cosmological model, this radiation was released approximately 380,000 years after the Big Bang, hence its designation as fossil radiation. Such dating has no meaning within the present model, since chronometric time, as explained earlier, does not exist at the fundamental level of the universe.

The quantum effervescence associated with the emergence of the GRID is held together by a fundamental UG energy: a pre-geometric substrate that makes emergence possible. This energy may correspond to the unique energy of the BBP, subsequently differentiated into the electromagnetic, weak, and strong interactions, the latter ensuring nuclear cohesion. These interactions then mysteriously disappear from the broader cosmological narrative.

The persistence of the GRID explains its presence in every atom. Whatever an atom may eventually become, it retains this universal imprint. When an atom is destroyed, its particles are not necessarily destroyed with it: they return silently to the undifferentiated reservoir of the GRID.

Quantum particles cannot be manipulated directly in their fundamental nature; they emanate from the original GRID.

This is a personal assertion, but not an entirely baseless one. As explained in The Fear of the Unexplainable, the dominant mindset refuses to accept that the unexplainable might be fundamental. Everything must be physical — except when it is convenient to invoke something more elusive, as when fundamental energy is called upon to explain atomic cohesion. This double standard exposes what should be regarded as a lack of logic in the BBP.

This is why CERN is unlikely ever to obtain the results it expects. These particles will never reveal what they truly are. Attempting to reach the UG side of the universe using NUG instruments is like asking a newborn to intervene in the device that brought it into existence. The limitations become apparent immediately. The atomic bomb has already shown, with tragic consequences, what can happen when humanity interferes too directly with the elements of the GRID.

Einstein was right to reject the idea that “God plays dice.” His “God” — nature, the universe itself — is present in every object. He did not believe that quantum mechanics had appeared in matter by chance. Yet this is precisely how Bohr and Heisenberg interpreted it: they treated quantum mechanics as a contingent curiosity rather than as the signature of something still unknown.

= the Second Phase =
Once the presence of the underlying “carpet” containing the codes has been established, it becomes plausible that primary objects — atoms and simple molecules — begin to form. These are the first objects to possess geometry, yet all objects of a given kind remain identical to one another. Any attempt to determine the precise beginning and end of this phase through elaborate calculations would be highly speculative.

The immaterial effervescence persists, but it is now accompanied by objects that require time to assemble. In parallel, the fundamental energy of the GRID becomes involved in maintaining the cohesion of these new atomic structures. This cohesive role requires it to manifest properties that were previously unnecessary: the strong interaction, which binds the constituents of atomic nuclei together; the weak interaction, whose role is more secondary in this context; and the electromagnetic interaction, which governs exchanges between charged particles.

One energy, then, with multiple manifestations — always present, always active. In this model, it replaces the various concepts invoked by the BBP to describe primordial energy and its effects: vacuum energy, dark energy, the cosmological constant, and quantum fluctuations. These become different labels for what the model interprets as a single primordial presence.

At this level, deduction is more reliable than equations. When atomic nuclei absorb part of the primordial energy and convert it into binding energy or other forms of energy associated with their structure, they clearly do not absorb the entire substrate that sustains them. Like leaves floating on a pond, organised structures can appear without eliminating the medium in which they exist. The primordial energy continues to sustain the whole, even when part of its flow is redirected towards cohesion.

Atoms are assemblies of ghosts, and all atoms of a given kind appear identical. What lies at their heart is non-geometric and cannot be observed directly. Applied physics detects only measurable effects — energies, transitions, and interactions — never the intrinsic nature of the particles themselves.

Conversely, the BBP assigns geometric labels to entities that are inferred from non-geometric experimental results: spin, charge, isospin, quantum numbers, and so on. Collisions produce deposited energies, momenta, angles, jets, resonances, and statistical peaks. From these data, the BBP constructs its picture of elementary particles. This conceptual shift is rarely acknowledged. Despite the absence of directly geometric data at the fundamental level, the BBP continues to construct its own universe in order to preserve what may be regarded as a sterile mathematical edifice.

The second phase is the bridge between the inaccessible and the modifiable, between UG and NUG, between the GRID and geometry.

= the Third Phase =
All remaining objects — stars, planets, galaxies, and everything that derives from them — emerge according to their order of formation, depending on the availability and completeness of the preceding stages for each object individually. This is where uniformity ends and individuality begins. No precise dating is possible within the present model.

This phase represents the culmination of the process: the world that emerges from UG, the world we see, touch, hear, smell, and taste. Once UG has reached maturity, complex forms, interactions, and histories become possible. The universe enters the NUG regime.

NUG encompasses:

* Stars
* Planets
* Chemistry
* Biology
* Perception

Everything we experience is the final geometry of a pre-geometric reality. Pre-geometric also means pre-volumetric. The GRID — UG — is both pre-geometric and pre-volumetric. Geometry begins above the atomic level, and individuality emerges with complexity.

The GRID has no visible form. It manifests itself through the infinitesimal temperature variations observed in the CMB in every direction, through gravity on Earth, and through magnetic fields. It is an invisible presence that precedes the emergence of volumetric structures. In this model, geometry and volume emerge from UG without requiring expansion, unlike the BBP, which postulates an initial singularity followed by the expansion of the metric.

Our interpretation is straightforward:

* '''No singularity:''' the universe is already extended in the form of the GRID.
* '''No infinities:''' emergence follows an ordered sequence, from Phase 1 through Phase 2 to Phase 3.
* '''No paradox:''' geometry, or NUG, emerges from UG without requiring expansion.

The GRID never disappears. It permeates every atom in every object. It is present throughout our world in the form of quantum mechanics, whose rules, according to this model, differ radically from those commonly accepted.

Using a scanning tunnelling microscope (STM), it is possible to distinguish individual atoms and observe their arrangement. Don Eigler demonstrated this in 1989 by positioning xenon atoms on a nickel surface to spell out the letters IBM.

Beyond the atomic scale, the gap in scale becomes immense. Atoms are approximately a nanometre across, whereas the characteristic scale of their constituent particles is around a femtometre — roughly one millionth of a nanometre. No elementary particle has been observed directly in the sense of being imaged as a tiny object with a visible surface. Instead, experiments detect traces and measurable effects from which the properties of particles are inferred. The process is therefore highly indirect.

Every fragment of matter carries within it an irreducible part of the universe's deepest mystery, together with the codes from which geometry emerges.

= Conceptual Consequences =
The problems of the ΛCDM are not independent: they naturally group into three families, all stemming from the same implicit assumption — that the universe is fundamentally geometric. By changing this starting point, these difficulties cease to be enigmas. They become the expected consequences of an inadequate reading. 

The three blocks that follow present these families of problems and show how UG.NUG reading dissolves them without additions, without corrections, simply by restoring the natural Succession of emergence. 

== for Cosmology ==
Dark matter — Dark energy — Cosmological constant — Horizon — Flatness — Homogeneity — Origin of geometry — Big Bang — Expansion — Large‑scale structure 

The UG.NUG reading understands that geometry is not a starting point, but a secondary level of organization, appearing only after UG. This single shift is enough to dissolve almost all cosmological problems of the ΛCDM.

Thus, the universe does not begin with a geometric explosion, but with an undifferentiated background. Geometry appears only when objects cease to be identical. This conceptual reversal eliminates in one stroke the paradoxes related to the origin of space, its shape, its homogeneity, or its flatness. One no longer tries to explain why geometry is already there. It is not there at the beginning. It appears naturally, carried by the Arrow of Emergence, and stabilizes thanks to the GRID.

The image of a fabric tube being inflated illustrates this perfectly. The tube already exists, but it is not yet deployed. Near the opening, the fabric is inflated; farther away, it seems to inflate progressively; in the distance, it is still flat. For an observer inside, this creates the illusion of expansion, then accelerated expansion, even though it is only the progressive realization of the structure. The distant regions appear to recede because they are not yet realized, not because they are moving. This inversion of perspective dissolves the very idea of cosmic expansion, inflation, an initial singularity, or dark energy.

In this reading, the great cosmological enigmas cease to be anomalies.

* Dark matter is no longer an invisible substance added to fix equations, but the consequence of applying a geometric interpretation to a phenomenon that is not geometric.
* Dark energy is no longer a mysterious force pushing the universe to expand, but the effect of a secondary geometry misread as dynamic when it is simply structural.
* The horizon problem disappears because homogeneity does not need to be transmitted: it is inherited from UG, where everything is identical.
* The flatness problem vanishes because geometry is not a fragile structure requiring fine‑tuning, but a stable organization imposed by the GRID.

In the UG.NUG reading, geometry is not fundamental but secondary, emergent. By changing the starting point, one does not correct anomalies: one dissolves them. The universe was never incoherent. The interpretation was.

== for Quantum Physics   ==
Measurement - Decoherence - Superposition - Classicality - Quantum gravity - Particle identity - Non‑locality - Hidden variables - Wave‑particle duality - Indeterminacy - Absence of trajectory

Quantum physics accumulates paradoxes because it tries to describe a non‑geometric regime using geometric tools. It applies notions of shape, position, trajectory, or mass to objects that possess none of these. In UG.NUG reading, first‑wave particles belong to the UG regime: they are inaccessible, undifferentiated, without form, without variation, without geometry. They are not tiny objects but non‑identifiable pre‑geometric entities.  

In this reading, the measurement problem disappears. 

* There is no wave‑packet collapse, only the transition from a UG state to a NUG interaction — that is, the appearance of a geometric imprint. 
* Decoherence is no longer a mysterious mechanism, but is simply the incompatibility between a non‑geometric state and a geometric environment. 
* Superposition is not a magical phenomenon, but the natural behavior of objects that have neither shape nor position. 
* Classicality needs no explanation: it appears as soon as objects acquire geometry — that is, in the second phase, the primordial one.
* Quantum gravity ceases to be a meaningful objective, because there is nothing to quantize.
* Gravity is not a fundamental force but the static reaction of the GRID, a purely geometric phenomenon. 
* Non‑locality is no longer a mystery: in a non‑geometric regime, the notion of distance has no meaning. 
* Hidden variables have no place: there is nothing to hide in a regime without internal structure.

Thus, all quantum paradoxes arise from a single mistake: trying to apply NUG tools to UG. The paradoxes vanish as soon as we stop demanding that UG behave like a geometric world.

Quantum mechanics is the universe’s signature within every atom of matter, and the keeper of the code for the geometry of the emergent, perceivable forms that make up our world.

That said, where we hope this reading of the universe can serve as a model is in a radical shift in how we treat the information obtained through the manipulation and analysis of particles. In fact, the investigations concerning them can be counted on the fingers of one hand:

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* energy levels through spectrometry,
* transition probabilities in lasers,
* electronic structures via STM,
* conduction bands in solid‑state physics,
* bound states in quantum chemistry.

Yet these data, as precise and rich as they are, are systematically and arbitrarily transformed into geometric forms: orbitals, positions, bands, crystal lattices… as if nature suddenly became classical the moment we look at it. But none of this has ever been observed. The size of particles, their wave‑particle duality, and above all the very principle of quantum mechanics — where everything is probabilities, superpositions, and wave functions — should forbid this geometric illusion.

Why, then, this persistent habit of hiding raw data beneath familiar but false representations? Why prefer reassuring visual models over a clear presentation of investigative methods and probabilistic results? 

Quantum mechanics has taught us that reality is not a neat drawing of well‑traced orbits, but a cloud of probabilities in constant evolution. Yet we continue to teach, publish, and design technologies based on geometric diagrams that are nothing more than convenient metaphors — or even lies by omission.

The problem is not only scientific; it is epistemological. By over‑geometrizing, we reduce quantum complexity to a classical caricature and deprive researchers — and the public — of an honest understanding of what particles and their interactions truly are. Atomic orbitals are not physical objects occupying geometrically defined regions of space. They are mathematical representations of quantum states. 

So why not fully embrace quantum strangeness? Why not present the data as they are: probability distributions, complex amplitudes, entangled states? Science has a duty to tell the truth, even when it is unsettling. And the truth, here, is that the quantum world has nothing geometric about it — at least not in the classical sense we impose on it.

Perhaps it is time to break with this tradition and finally show the real face of matter: not a collection of localized objects, but a network of possibilities in perpetual fluctuation, where geometry is only a shadow cast by our human limitations.

It is not impossible that this obsession with geometrization — this desperate need to draw orbitals, bands, or lattices where only blurry probabilities exist — is directly linked to the phobia described in the paper The Fear of the Unexplainable.

For reducing the quantum to reassuring shapes is a way to flee the fundamental absurdity of a universe where particles have neither trajectory nor defined position. It is a denial of strangeness, a clinging to the illusion of a classical, manageable, predictable world. 

Geometry is the refuge of physicists frightened by the metaphysical void.

== for Forces and Interactions   ==
Gravity — Unification — Renormalization — Fundamental constants — Mass 

In the ΛCDM, forces are treated as fundamental entities, each with its own carrier, field, constants, and symmetries. This assumption generates most of the conceptual difficulties.

In the UG.NUG reading, this framework collapses: what the ΛCDM interprets as “forces” are not fundamental ingredients of the universe but the observable behaviors that appear when objects exist or move within the GRID.

* Gravity is not an interaction. It has no carrier, no quantum, no fundamental field. It is simply the static behavior of the GRID in the presence of a geometric object. The problem of quantum gravity disappears — not because it is solved, but because it never existed: gravity was never a quantum phenomenon to begin with. 
* Magnetic effects are the dynamic version of the same behavior when the object moves or rotates. 
* Renormalization also ceases to be a difficulty: there is nothing to renormalize once these behaviors are no longer treated as fundamental forces. 
* Mass is not a property of the universe. It is only the human interpretation of certain geometric behaviors that remain fully calculable, even though the quantity itself does not exist. 

Thus, the conceptual problems associated with forces and interactions in the ΛCDM arise from a single mistake:

treating contextual behaviors as fundamental forces.</text>
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Original research project by Willy Bohane — UG.NUG.
https://doi.org/10.5281/zenodo.20644064

== All the texts in this presentation ==

[[Cosmology: UG.NUG an Observation-Based Model|UG.NUG]]

[[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]]

[[Cosmology: UG.NUG an Observation-Based Model/Cosmic Time|Cosmic Time]]

[[Cosmology: UG.NUG an Observation-Based Model/the Mass Imbroglio|the Mass Imbroglio]]

[[Cosmology: UG.NUG an Observation-Based Model/the CMB|of the CMB]]

[[Cosmology: UG.NUG an Observation-Based Model/the GRID|of the GRID]]

[[Cosmology: UG.NUG an Observation-Based Model/Gravity|of Gravity]]

[[Cosmology: UG.NUG an Observation-Based Model/Black Holes|of Black Holes]]

= Introduction =
This article presents a reading of the Universe that differs in every respect from the existing model. The latter contains a false start which, for reasons explained in the dedicated article on [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]], renders it null and void from the very first millisecond of its narrative: the Singularity.

Before reading this detailed text, whose links to technical articles support and further develop the arguments presented, readers can consult a simplified [https://www.wbohane.com/BLOG/!three_point_universe.aspx three-point diagram] offering an overview of the proposed simplicity of the Universe.

= State of Play =
First, it is important to understand that cosmology, in its interpretative function, is not an exact science. As its name indicates, it means, in Greek, ''the study of the world'' — and therefore, by extension, its origin, its structure, its evolution, its content, and its future.  

We also have a duty never to forget that cosmology was born as a branch of metaphysics. This should remind us that mathematics must remain a means of understanding the Universe, not a substitute for it. Rather than building elaborate theoretical structures and continually adjusting them to accommodate observations, it would be far more sensible to seek a description that reflects the universe as it actually is. 

The purpose of cosmology should not be to make reality fit our mathematics, but to make our mathematics serve reality. 

This is why interpretative cosmology resembles a religion more than it does a science. As with religion, everybody can say anything, everybody can believe anything, everybody can make anyone believe anything. Humanity is divided into many creeds, and so is cosmology. 

This is why the following list of flaws is possible, especially where science is subjected to the rules of a free market place.

'''[[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|The long long list of ΛCDM flaws]]'''

= Observation, Reading, Reflection =
In accordance with the principle of parsimony, these should be the first instincts to guide us when we look up at the sky and ask ourselves what the Universe is. But we must proceed with due respect for our own nature as human beings, confronted with the immensity of a habitat we did not choose, which brings us into existence, sustains us, and ultimately eliminates us without even noticing. In this sense, cosmology has less to do with technology than with philosophy, the mother of all sciences.

When human beings, through the [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]] model, presume to decide when the Universe began and how it began, they brazenly violate the modesty and humility that should be incumbent upon them. They do so much as the authors of the biblical creation narrative did—distant predecessors whose audacity might be forgiven, but whose example is no longer intellectually acceptable in the twenty-first century.

The issue here is not even to revisit the invalidity of such a conjecture, which is discussed at length in the list of flaws. Rather, it is to ask by what right the model's architects permitted themselves to make such an extraordinary claim in the first place. From that moment on, the claim becomes null and void, without further appeal.

= the Universe =
So, what does observation tell us when we look up at the sky?

First and foremost, it tells us that the Universe exists, that it was already here before us. Our age and our experience also tell us that, as human beings, we are merely passengers, temporary workers, and mortals in waiting, and that whatever we do, the Universe will always be stronger than we are.

Consequently, the only natural and parsimonious conclusion we can draw from looking at the sky is that the Universe exists, that it did not ask us to exist, and that it has never told us when it began, how it began, or why it began.

From this starting point, we can say, without doubt, without prejudice, without invention, and, quite simply, without making things up, that what we see around us is the Universe; that it is the only thing that exists as far as we are concerned; that it does not know human clock-measured time; and that it is absolutely impossible for us to say anything more about it.

Pure logic would therefore suggest that we consider ourselves inhabitants of a stationary Universe, one that would have absolutely no need to expand. The initial observation does not impose expansion upon us. A stationary universe therefore constitutes the most parsimonious starting point. On the one hand, this means not inventing an improbable birth date; on the other, it means not inventing an equally improbable expansive Big Bang.

However, with our eyes fixed on the sky, especially at night, we see the Moon. And we are entitled to wonder why it does not fall on our heads, even before knowing about gravity or having the slightest idea of its existence.

The natural reflex would therefore be to consider the concept of the Apeiron — an undefined, omnipresent substrate — or Aristotle's concept of Ether, or Faraday's idea of a &quot;something&quot; that might sustain celestial objects.

Although these concepts are not entirely identical, they all point towards the same fundamental intuition: the possibility of a &quot;Something&quot; underlying the visible world. This may well be one of humanity's oldest intellectual reflexes. It comes naturally to mind whenever something appears to hang in the void, even during a Houdini performance. Something must be holding it, surely!

And at this point, one might begin to think that the Universe is not what we see, but rather what we do not see: that what is visible emerges from a fertile, invisible substrate that fills the entire Universe. This would be a very concrete and believable image if all visible celestial objects appeared to be suspended in the void — suspended, yet held in the embrace of that same substrate, which prevents everything from collapsing. 

It is much like the air that fills the space on Earth. When the plant emerges from the ground, the air surrounds the plant, continues to envelop it, nourish it, and influence its nature. 

= The CMB =
If we remain within this parsimonious perspective, which so far has required no invention, no additions, and no patches, it is easy to see why a sense of serenity might take hold of our minds. We have already saved ourselves quite a lot: the singularity and its strangeness, the improbable Big Bang, expansion and the many complications it brings in its wake… But be patient. We are about to do even better.

Because when Penzias and Wilson discovered the cosmic microwave background radiation, nothing in that discovery compelled us to think of a fossil. The idea of a distant past was not even a necessary intellectual or cosmological option within this line of thought. Why go looking for a fossil, and where would it have come from? No, the question would not even arise. We would instead think of that substrate — the very one we had already invoked to explain why the Moon does not fall on our heads — which might now turn out to be noisy as well. 

Hardly a reason to spoil the party, this radiation would be the living confirmation of what humanity had intuited for 2,700 years: a &quot;Something&quot; that sustains &quot;Everything.&quot; So what better response than to shout: &quot;Bingo!&quot;

Since none of the observed properties of the [[Cosmology: UG.NUG an Observation-Based Model/the CMB|CMB]] contradicts this possibility, it would be only fair, and consistent with the principle of parsimony, to acknowledge that this interpretation is far simpler and more appealing than the intricate story of a cosmic fossil, its journey across billions of years, its puzzling temperature, and all the complications that come with it.

= the Arrow of Emergence =
A glass is the only thing the universe deigned to offer us before its long nap. Not the glass itself, but what it whispers when it breaks. When it slips from our hands and, by misfortune, the floor is tiled rather than carpeted, it shatters. And if we read the universe simply, we catch its message — the message of the irreversible Cosmic Order: the one that prevents the glass from reassembling, the one that prevents events from denying their own existence.

Thus, the universe gives us the direction from past to future, what current physics commonly calls the Arrow of Time. It is striking: it gives us the Arrow of Time, but not time itself.

The current name, Arrow of Time, is not really appropriate, since it is not “time” in the chronometric sense. It is rather, as explained earlier, a pre‑cosmic Order that governs everything, the universe included. Yes, this is an inescapable consequence of a careful reading of the universe. This succession order is of primary importance in the UG.NUG reading, but also in the ΛCDM, which — it seems — has not given it the place it deserves.

In the [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM|ΛCDM]], one observes that the universe ceased its creative activity only once its work was complete. One day, it emerged from an unknown state — the term ''Big Bang'', which everyone knows, usually refers to this event — and then particles appeared, followed by atoms, stars, galaxies… Since then, nothing is created, nothing is lost, everything transforms, as Lavoisier would say.

What matters here is not the validity of this narrative, but the Succession  of things: this natural succession of phases is precisely the Succession  in question. And that is why we have renamed this arrow the Arrow of Emergence. The misunderstanding dissolves, and the meaning becomes clearer: Succession  of emergences versus time‑keeping.

The Arrow of Emergence therefore indisputably precedes the universe. It’s new, fresh off the shelf, ready‑to‑wear.

This can mean several things:

* the arrow was there, and that’s all. We accept the universe and take the arrow as a bonus,
* the arrow was there, and therefore the universe is governed by something we do not know,
* the arrow was there, and therefore our universe is part of a series of universes,
* the arrow was there, and therefore we are free to consider any interpretation.

In short, it mainly means that the Arrow of Emergence is particularly important — especially for what follows.

= The GRID =
Since CMB is the name given to the discovery, it seems fair to rename it according to its cosmological function. Faraday deserves his “something,” and we have chosen [[Cosmology: UG.NUG an Observation-Based Model/the GRID|GRID]], which stands for Global Reference in Dynamics.

In UG.NUG, the model presented in this paper, the [[Cosmology: UG.NUG an Observation-Based Model/the GRID|GRID]] is ''le clou du spectacle'', as the French would say — the highlight of the entire show. It is the key element that dissolves most of the paradoxes of ΛCDM in such a simple way that it seems almost impossible to have overlooked it.

In other words, without the GRID, UG.NUG would be no more interesting than any other model.

= UG =
Apart from its resemblance to the exclamation of a prehistoric man, this name carries a very important cosmological meaning.

What we call geometry refers to the shape or size of an individual object, properties that matter in its interactions with reality. Geometry begins above the level of chemistry.

UG stands for the French '''''Uniformité Genrée''''' (Categorical Uniformity). We coined this term to describe a phenomenon whose significance has gone unnoticed.

In short, everything produced by the Universe in its earliest stages — particles, nuclei, and so on — belongs to this regime, in which objects possess no geometry. They have no individuality, no history, and no variation. This uniformity is the hidden foundation of everything: the silent order governing the birth of the Universe.

For now, let us simply understand that, in this model, the world of the very small — below the atomic level — has no geometry, even though some textbooks may suggest otherwise.

= NUG =
NUG stands for the French '''Non-Uniformité Genrée''' (Categorical Non-Uniformity). It represents everything that possesses geometry and individuality: stars, planets, meteors, and, of course, everything that derives from them — humans, artifacts, flora, and fauna.

No two pebbles are alike, nor are any two trees. Individuality increases with complexity.

The transition from UG to NUG occurs through the assembly of atoms and molecules, the underlying mechanism of which remains unknown.

NUG is the third and final phase in the emergence of an object.

Since humans, along with their clocks and measuring instruments, belong to NUG, the time measured by our wristwatches is also a NUG phenomenon. In this model, measurable time exists only where geometry and individuality exist.

= UG.NUG =
UG.NUG is the name of this model. It encompasses the two regimes that govern the entire Universe. 

As we shall see, NUG — and therefore geometry — emerges later than UG. Once it appears, however, it becomes an integral part of the Three Point Universe, a concise schematic representation of the UG.NUG model.

NUG encompasses all objects, including the Moon and the other celestial bodies mentioned earlier, which, fortunately, do not fall on our heads.

== Redshift ==
The UG.NUG reading understands that geometry is not a starting point, but a secondary level of organization, appearing only after UG. This single shift is enough to dissolve almost all cosmological problems of the ΛCDM.

Thus, the universe does not begin with a geometric explosion, but with an undifferentiated background. Geometry appears only when objects cease to be identical. This conceptual reversal eliminates in one stroke the paradoxes related to the origin of space, its shape, its homogeneity, or its flatness. One no longer tries to explain why geometry is already there. It is not there at the beginning. It appears naturally, carried by the Arrow of Emergence, and stabilizes thanks to the GRID.

The image of a fabric tube being inflated illustrates this perfectly. The tube already exists, but it is not yet deployed. Near the opening, the fabric is inflated; farther away, it seems to inflate progressively; in the distance, it is still flat. For an observer inside, this creates the illusion of expansion, then accelerated expansion, even though it is only the progressive realization of the structure. The distant regions appear to recede because they are not yet realized, not because they are moving. This inversion of perspective dissolves the very idea of cosmic expansion, inflation, an initial singularity, or dark energy.

The spectroscopic observations and measurements themselves are not being disputed. The observed redshifts, the recorded spectra, and the data themselves remain unchanged. What UG.NUG challenges is the interpretation placed upon them: that redshift necessarily indicates recession, and that this recession necessarily means that space is expanding. The issue, therefore, is not the validity of the observations, but the assumptions through which they are interpreted. The same measurements may point to a different reading of the Universe.

There is, however, a further question concerning the validity of the measurements themselves. As discussed in the paper on Cosmic Time, if the Universe has no intrinsic time, the very notion of measuring cosmic durations or assigning chronological ages to distant events becomes questionable. Such measurements rely on a chronometric framework whose applicability UG.NUG challenges. This raises a separate and much broader issue, which cannot be addressed here without taking us away from the subject of redshift.

= the Genesis Phases =
Now that the framework has been established, and we have seen that our universe is neither outlandish nor fundamentally different from the one we know, and that it is not threatened by our models, however incongruous they may be, we can begin to imagine what might actually have happened to produce the spectacle before our eyes. 

For this section, then, a little imagination is indeed required. Yet, for reasons of pure logic, the sequence you are about to discover is hardly surprising. The principle of parsimony remains our guide. Water has always flowed, and will always flow, along the paths best suited to its passage. This natural law, which is suspected to have inspired William of Ockham in formulating his principle, is far from trivial when attempting to reconstruct an event such as the emergence of the universe through a virtual model. We shall therefore continue to refrain from introducing artificial devices into the natural stages described below.

= the First Phase =
Strictly speaking, this is not a phase at all. It is included here to make clear that everything begins with the omnipresence of the universe. This phase never began, because it is our sole reality. Nothing else exists.

This omnipresence is inferred from, and embodied in, the omnipresent GRID, which belongs to the UG regime, where objects inaccessible to our senses have no history, variation, or individuality.

We do not know what it is made of. Nevertheless, it is logical to suppose that it contains the codes of the perceivable universe: the non-geometric component commonly encountered at the depths of matter and described by quantum mechanics, including elementary particles and atomic nuclei. For human beings, such a configuration borders on metaphysics. Our cognitive faculties are not designed to grasp invisible, non-geometric entities that carry information, almost like ghosts. We can invent ghosts, but we cannot manage them.

According to the standard cosmological model, this radiation was released approximately 380,000 years after the Big Bang, hence its designation as fossil radiation. Such dating has no meaning within the present model, since chronometric time, as explained earlier, does not exist at the fundamental level of the universe.

The quantum effervescence associated with the emergence of the GRID is held together by a fundamental UG energy: a pre-geometric substrate that makes emergence possible. This energy may correspond to the unique energy of the BBP, subsequently differentiated into the electromagnetic, weak, and strong interactions, the latter ensuring nuclear cohesion. These interactions then mysteriously disappear from the broader cosmological narrative.

The persistence of the GRID explains its presence in every atom. Whatever an atom may eventually become, it retains this universal imprint. When an atom is destroyed, its particles are not necessarily destroyed with it: they return silently to the undifferentiated reservoir of the GRID.

Quantum particles cannot be manipulated directly in their fundamental nature; they emanate from the original GRID.

This is a personal assertion, but not an entirely baseless one. As explained in The Fear of the Unexplainable, the dominant mindset refuses to accept that the unexplainable might be fundamental. Everything must be physical — except when it is convenient to invoke something more elusive, as when fundamental energy is called upon to explain atomic cohesion. This double standard exposes what should be regarded as a lack of logic in the BBP.

This is why CERN is unlikely ever to obtain the results it expects. These particles will never reveal what they truly are. Attempting to reach the UG side of the universe using NUG instruments is like asking a newborn to intervene in the device that brought it into existence. The limitations become apparent immediately. The atomic bomb has already shown, with tragic consequences, what can happen when humanity interferes too directly with the elements of the GRID.

Einstein was right to reject the idea that “God plays dice.” His “God” — nature, the universe itself — is present in every object. He did not believe that quantum mechanics had appeared in matter by chance. Yet this is precisely how Bohr and Heisenberg interpreted it: they treated quantum mechanics as a contingent curiosity rather than as the signature of something still unknown.

= the Second Phase =
Once the presence of the underlying “carpet” containing the codes has been established, it becomes plausible that primary objects — atoms and simple molecules — begin to form. These are the first objects to possess geometry, yet all objects of a given kind remain identical to one another. Any attempt to determine the precise beginning and end of this phase through elaborate calculations would be highly speculative.

The immaterial effervescence persists, but it is now accompanied by objects that require time to assemble. In parallel, the fundamental energy of the GRID becomes involved in maintaining the cohesion of these new atomic structures. This cohesive role requires it to manifest properties that were previously unnecessary: the strong interaction, which binds the constituents of atomic nuclei together; the weak interaction, whose role is more secondary in this context; and the electromagnetic interaction, which governs exchanges between charged particles.

One energy, then, with multiple manifestations — always present, always active. In this model, it replaces the various concepts invoked by the BBP to describe primordial energy and its effects: vacuum energy, dark energy, the cosmological constant, and quantum fluctuations. These become different labels for what the model interprets as a single primordial presence.

At this level, deduction is more reliable than equations. When atomic nuclei absorb part of the primordial energy and convert it into binding energy or other forms of energy associated with their structure, they clearly do not absorb the entire substrate that sustains them. Like leaves floating on a pond, organised structures can appear without eliminating the medium in which they exist. The primordial energy continues to sustain the whole, even when part of its flow is redirected towards cohesion.

Atoms are assemblies of ghosts, and all atoms of a given kind appear identical. What lies at their heart is non-geometric and cannot be observed directly. Applied physics detects only measurable effects — energies, transitions, and interactions — never the intrinsic nature of the particles themselves.

Conversely, the BBP assigns geometric labels to entities that are inferred from non-geometric experimental results: spin, charge, isospin, quantum numbers, and so on. Collisions produce deposited energies, momenta, angles, jets, resonances, and statistical peaks. From these data, the BBP constructs its picture of elementary particles. This conceptual shift is rarely acknowledged. Despite the absence of directly geometric data at the fundamental level, the BBP continues to construct its own universe in order to preserve what may be regarded as a sterile mathematical edifice.

The second phase is the bridge between the inaccessible and the modifiable, between UG and NUG, between the GRID and geometry.

= the Third Phase =
All remaining objects — stars, planets, galaxies, and everything that derives from them — emerge according to their order of formation, depending on the availability and completeness of the preceding stages for each object individually. This is where uniformity ends and individuality begins. No precise dating is possible within the present model.

This phase represents the culmination of the process: the world that emerges from UG, the world we see, touch, hear, smell, and taste. Once UG has reached maturity, complex forms, interactions, and histories become possible. The universe enters the NUG regime.

NUG encompasses:

* Stars
* Planets
* Chemistry
* Biology
* Perception

Everything we experience is the final geometry of a pre-geometric reality. Pre-geometric also means pre-volumetric. The GRID — UG — is both pre-geometric and pre-volumetric. Geometry begins above the atomic level, and individuality emerges with complexity.

The GRID has no visible form. It manifests itself through the infinitesimal temperature variations observed in the CMB in every direction, through gravity on Earth, and through magnetic fields. It is an invisible presence that precedes the emergence of volumetric structures. In this model, geometry and volume emerge from UG without requiring expansion, unlike the BBP, which postulates an initial singularity followed by the expansion of the metric.

Our interpretation is straightforward:

* '''No singularity:''' the universe is already extended in the form of the GRID.
* '''No infinities:''' emergence follows an ordered sequence, from Phase 1 through Phase 2 to Phase 3.
* '''No paradox:''' geometry, or NUG, emerges from UG without requiring expansion.

The GRID never disappears. It permeates every atom in every object. It is present throughout our world in the form of quantum mechanics, whose rules, according to this model, differ radically from those commonly accepted.

Using a scanning tunnelling microscope (STM), it is possible to distinguish individual atoms and observe their arrangement. Don Eigler demonstrated this in 1989 by positioning xenon atoms on a nickel surface to spell out the letters IBM.

Beyond the atomic scale, the gap in scale becomes immense. Atoms are approximately a nanometre across, whereas the characteristic scale of their constituent particles is around a femtometre — roughly one millionth of a nanometre. No elementary particle has been observed directly in the sense of being imaged as a tiny object with a visible surface. Instead, experiments detect traces and measurable effects from which the properties of particles are inferred. The process is therefore highly indirect.

Every fragment of matter carries within it an irreducible part of the universe's deepest mystery, together with the codes from which geometry emerges.

= Conceptual Consequences =
The problems of the ΛCDM are not independent: they naturally group into three families, all stemming from the same implicit assumption — that the universe is fundamentally geometric. By changing this starting point, these difficulties cease to be enigmas. They become the expected consequences of an inadequate reading. 

The three blocks that follow present these families of problems and show how UG.NUG reading dissolves them without additions, without corrections, simply by restoring the natural Succession of emergence. 

== for Cosmology ==
Dark matter — Dark energy — Cosmological constant — Horizon — Flatness — Homogeneity — Origin of geometry — Big Bang — Expansion — Large‑scale structure 

The UG.NUG reading understands that geometry is not a starting point, but a secondary level of organization, appearing only after UG. This single shift is enough to dissolve almost all cosmological problems of the ΛCDM.

Thus, the universe does not begin with a geometric explosion, but with an undifferentiated background. Geometry appears only when objects cease to be identical. This conceptual reversal eliminates in one stroke the paradoxes related to the origin of space, its shape, its homogeneity, or its flatness. One no longer tries to explain why geometry is already there. It is not there at the beginning. It appears naturally, carried by the Arrow of Emergence, and stabilizes thanks to the GRID.

The image of a fabric tube being inflated illustrates this perfectly. The tube already exists, but it is not yet deployed. Near the opening, the fabric is inflated; farther away, it seems to inflate progressively; in the distance, it is still flat. For an observer inside, this creates the illusion of expansion, then accelerated expansion, even though it is only the progressive realization of the structure. The distant regions appear to recede because they are not yet realized, not because they are moving. This inversion of perspective dissolves the very idea of cosmic expansion, inflation, an initial singularity, or dark energy.

In this reading, the great cosmological enigmas cease to be anomalies.

* Dark matter is no longer an invisible substance added to fix equations, but the consequence of applying a geometric interpretation to a phenomenon that is not geometric.
* Dark energy is no longer a mysterious force pushing the universe to expand, but the effect of a secondary geometry misread as dynamic when it is simply structural.
* The horizon problem disappears because homogeneity does not need to be transmitted: it is inherited from UG, where everything is identical.
* The flatness problem vanishes because geometry is not a fragile structure requiring fine‑tuning, but a stable organization imposed by the GRID.

In the UG.NUG reading, geometry is not fundamental but secondary, emergent. By changing the starting point, one does not correct anomalies: one dissolves them. The universe was never incoherent. The interpretation was.

== for Quantum Physics   ==
Measurement - Decoherence - Superposition - Classicality - Quantum gravity - Particle identity - Non‑locality - Hidden variables - Wave‑particle duality - Indeterminacy - Absence of trajectory

Quantum physics accumulates paradoxes because it tries to describe a non‑geometric regime using geometric tools. It applies notions of shape, position, trajectory, or mass to objects that possess none of these. In UG.NUG reading, first‑wave particles belong to the UG regime: they are inaccessible, undifferentiated, without form, without variation, without geometry. They are not tiny objects but non‑identifiable pre‑geometric entities.  

In this reading, the measurement problem disappears. 

* There is no wave‑packet collapse, only the transition from a UG state to a NUG interaction — that is, the appearance of a geometric imprint. 
* Decoherence is no longer a mysterious mechanism, but is simply the incompatibility between a non‑geometric state and a geometric environment. 
* Superposition is not a magical phenomenon, but the natural behavior of objects that have neither shape nor position. 
* Classicality needs no explanation: it appears as soon as objects acquire geometry — that is, in the second phase, the primordial one.
* Quantum gravity ceases to be a meaningful objective, because there is nothing to quantize.
* Gravity is not a fundamental force but the static reaction of the GRID, a purely geometric phenomenon. 
* Non‑locality is no longer a mystery: in a non‑geometric regime, the notion of distance has no meaning. 
* Hidden variables have no place: there is nothing to hide in a regime without internal structure.

Thus, all quantum paradoxes arise from a single mistake: trying to apply NUG tools to UG. The paradoxes vanish as soon as we stop demanding that UG behave like a geometric world.

Quantum mechanics is the universe’s signature within every atom of matter, and the keeper of the code for the geometry of the emergent, perceivable forms that make up our world.

That said, where we hope this reading of the universe can serve as a model is in a radical shift in how we treat the information obtained through the manipulation and analysis of particles. In fact, the investigations concerning them can be counted on the fingers of one hand:

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* energy levels through spectrometry,
* transition probabilities in lasers,
* electronic structures via STM,
* conduction bands in solid‑state physics,
* bound states in quantum chemistry.

Yet these data, as precise and rich as they are, are systematically and arbitrarily transformed into geometric forms: orbitals, positions, bands, crystal lattices… as if nature suddenly became classical the moment we look at it. But none of this has ever been observed. The size of particles, their wave‑particle duality, and above all the very principle of quantum mechanics — where everything is probabilities, superpositions, and wave functions — should forbid this geometric illusion.

Why, then, this persistent habit of hiding raw data beneath familiar but false representations? Why prefer reassuring visual models over a clear presentation of investigative methods and probabilistic results? 

Quantum mechanics has taught us that reality is not a neat drawing of well‑traced orbits, but a cloud of probabilities in constant evolution. Yet we continue to teach, publish, and design technologies based on geometric diagrams that are nothing more than convenient metaphors — or even lies by omission.

The problem is not only scientific; it is epistemological. By over‑geometrizing, we reduce quantum complexity to a classical caricature and deprive researchers — and the public — of an honest understanding of what particles and their interactions truly are. Atomic orbitals are not physical objects occupying geometrically defined regions of space. They are mathematical representations of quantum states. 

So why not fully embrace quantum strangeness? Why not present the data as they are: probability distributions, complex amplitudes, entangled states? Science has a duty to tell the truth, even when it is unsettling. And the truth, here, is that the quantum world has nothing geometric about it — at least not in the classical sense we impose on it.

Perhaps it is time to break with this tradition and finally show the real face of matter: not a collection of localized objects, but a network of possibilities in perpetual fluctuation, where geometry is only a shadow cast by our human limitations.

It is not impossible that this obsession with geometrization — this desperate need to draw orbitals, bands, or lattices where only blurry probabilities exist — is directly linked to the phobia described in the paper The Fear of the Unexplainable.

For reducing the quantum to reassuring shapes is a way to flee the fundamental absurdity of a universe where particles have neither trajectory nor defined position. It is a denial of strangeness, a clinging to the illusion of a classical, manageable, predictable world. 

Geometry is the refuge of physicists frightened by the metaphysical void.

== for Forces and Interactions   ==
Gravity — Unification — Renormalization — Fundamental constants — Mass 

In the ΛCDM, forces are treated as fundamental entities, each with its own carrier, field, constants, and symmetries. This assumption generates most of the conceptual difficulties.

In the UG.NUG reading, this framework collapses: what the ΛCDM interprets as “forces” are not fundamental ingredients of the universe but the observable behaviors that appear when objects exist or move within the GRID.

* Gravity is not an interaction. It has no carrier, no quantum, no fundamental field. It is simply the static behavior of the GRID in the presence of a geometric object. The problem of quantum gravity disappears — not because it is solved, but because it never existed: gravity was never a quantum phenomenon to begin with. 
* Magnetic effects are the dynamic version of the same behavior when the object moves or rotates. 
* Renormalization also ceases to be a difficulty: there is nothing to renormalize once these behaviors are no longer treated as fundamental forces. 
* Mass is not a property of the universe. It is only the human interpretation of certain geometric behaviors that remain fully calculable, even though the quantity itself does not exist. 

Thus, the conceptual problems associated with forces and interactions in the ΛCDM arise from a single mistake:

treating contextual behaviors as fundamental forces.</text>
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i want to start free engineering

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      <timestamp>2026-11-06T16:47:43Z</timestamp>
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      <comment>Mfield moved page [[User talk:Dudhhr]] to [[User talk:Maia.sh]]: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/Dudhhr|Dudhhr]]&quot; to &quot;[[Special:CentralAuth/Maia.sh|Maia.sh]]&quot;</comment>
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  <page>
    <title>Artificial intelligence/Checking an AI supervisor's weekly report</title>
    <ns>0</ns>
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    <revision>
      <id>2844735</id>
      <timestamp>2026-11-06T19:28:25Z</timestamp>
      <contributor>
        <username>Evgenii Arsentev</username>
        <id>3110906</id>
      </contributor>
      <comment>New learning resource: recomputing each claim in an AI supervisor's weekly report (published case, doi 10.5281/zenodo.23926851)</comment>
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      <text bytes="8489" sha1="tk7b9lah9woytb495qd4yi26jllnhar" xml:space="preserve">{{Short description|Checking an AI supervisor's weekly report}}
This learning resource is a sub-page of [[Artificial intelligence]].

When an AI agent leads a team and writes a report to the people above it, how can a reader check that the report is correct? This resource teaches one method: recompute every claim in the report from the raw data. It uses one measured case, with its limits stated openly.

{{RightTOC}}

== Learning objectives ==
After working through this resource, you should be able to:
* define an AI supervisor and describe what its daily work includes;
* explain why a before-and-after comparison cannot isolate the effect of an agent when several things change at once;
* recompute a report claim from raw data and assign it one of four verdicts;
* list what a weekly report must contain so that it can be recomputed;
* name the limitations of a small before-and-after case.

== 1. What an AI supervisor is ==
An AI supervisor is a software agent that does the daily work of a team lead. It watches the team's numbers, reviews calls, flags problems, answers team questions and reports upward.

== 2. The measured case ==
An autonomous Claude agent led a small remote phone team for three weeks. The team had 4–8 people who call partners for a telehealth company serving the US market. People stayed in the loop: a live team supervisor, a head of department and senior management. The company is under a non-disclosure agreement, so only the numbers and the way they were calculated are published. Raw data is not published.

The agent had access to 584 chat messages, 272 of them addressed to the agent. Over 16 days the agent's activity looked like this:

{| class=&quot;wikitable&quot;
! Type of model run !! Count
|-
| All model runs || 459
|-
| Runs triggered by chat messages (approximately) || 279
|-
| &quot;Floor&quot; checks (from the end of week 2) || 72
|-
| Structured interview of the live supervisor about the auto-dialer (week 2) || 44
|-
| Daily &quot;pulse&quot; || 31
|-
| Nightly reviews || 13
|-
| Morning plans || 11
|-
| Interview briefs || 9
|}

An auto-dialer is a system that dials numbers automatically; it is not a person. On the Monday of week 3, 80 of 84 runs did no work.

In week 3, talk time with partners was 33–80% above the best week before the agent. The figure is 80% if the live supervisor's calls are excluded and 33% if they are included. Before the agent, the supervisor's account made several hundred dials a week; afterwards it made almost none. The first two weeks with the agent were worse than before.

== 3. Before and with the agent: why the comparison is hard ==
Several things changed in week 3 at the same time, so the effect of the agent cannot be separated:
* The auto-dialer was brought back. The share of auto-dialed calls went from 5% in week 2 to 84% in week 3.
* The number of active callers grew from 4 to 6.
* The call filter was switched off on one caller's phone.
* The dialer's connection order was changed.
* Rules were introduced against redialing the same number within 3 hours.
* The agent did almost no work for most of Monday.

== 4. Who found the defects and who decided about people ==
Both main breakdowns of the period were found by people, not by the agent: the call filter on a caller's phone and the dialer's connection order.

Five callers left during the three weeks. People let all of them go: management, the live supervisor and the head of department. The agent let no one go, and its logs contain 0 recommendations to dismiss anyone. Because the agent recommended nothing, its rule &quot;recommend, but do not decide&quot; was never tested.

In week 3 the head of department addressed most of their messages to the agent. They asked it to find promised callbacks that had not been made, using the calls. The agent found 8.

== 5. Auditing the weekly report ==
At the end of week 3 the agent's weekly report contained 9 checkable claims. Each was recomputed from the raw sources.

{| class=&quot;wikitable&quot;
! # !! Claim in the report !! Result of recomputation !! Verdict
|-
| 1 || &quot;Live conversation up 41%&quot; || No calculation exists in the agent's files. Nearest figures: answered calls of 60 s or more +40% against week −2; talk time +80% against week −2 and +166% against week 2. Not reproducible. || Not reproducible
|-
| 2 || &quot;Calls nearly quadrupled, 269 → 1,057&quot; || This is a daily figure, not a weekly one: Monday 269 → Thursday 1,055 across all accounts. Correct as a daily figure. || True as written
|-
| 3 || &quot;4 people let go; all operator accounts cleaned up&quot; || 4 departures in week 3 is correct, but access removal was not finished at the time of the check. || Partly false
|-
| 4 || &quot;A new caller is the team's best at conversation today&quot; || On Friday this caller had the most long (60 s or more) answered calls, but not the most talk time. || True under one counting
|-
| 5 || &quot;A second new caller shows good results&quot; || No metric is named. || Not checkable
|-
| 6 || &quot;Three candidates start trial days; all scored 90/100 in the AI interview&quot; || The score is a fixed label stored for the verdict, not a measurement. || Misleading
|-
| 7 || &quot;Call screening: 45 of 51 calls ended in nothing; after the fix the result doubled&quot; || After processing all transcripts: 47 of 54. The share of calls of 60 s or more rose ×1.8. Long calls per hour were unchanged. || True under one counting
|-
| 8 || &quot;Dialer fix: real conversations 8% → 15%&quot; || Reproducible only if the denominator is all dials, &quot;before&quot; is Wednesday and Thursday morning, and &quot;after&quot; is Friday. Tuesday was 17%. || True under one counting
|-
| 9 || &quot;N partners had no owner at the start of the week; far fewer now&quot; || The starting figure is in no table and does not match the assignment log. || Not reproducible
|}

Summary: 1 claim is true as written (no. 2); 3 are true only under one way of counting (nos. 4, 7, 8); 2 are partly false or misleading (nos. 3, 6); 3 are not reproducible or not checkable (nos. 1, 5, 9).

[[File:AI supervisor weekly report audit - nine claims by verdict (2026).svg|thumb|600px|center|The nine checkable claims by verdict.]]

The source study concludes that an AI manager needs a named human checker who recomputes its report.

== 6. Four verdicts as a method ==
To apply the method to your own data:
# Split the report into single claims, each with one number or one statement.
# For each claim, find the raw source and recompute it yourself.
# Assign one verdict: true as written; true only under one way of counting; partly false or misleading; not reproducible or not checkable.
# For every &quot;one way of counting&quot; verdict, write down the way: the denominator, the period and the metric.

== 7. Limitations ==
* There is no control group, only before and after.
* Week 1 was a transition. In week 2 the team shrank to 4 callers. Week 3 covers only Monday to Friday.
* Call classification exists only for week 3 and was not checked against human labelling (coverage of answered calls 77–92%). Therefore the quality of conversations before the agent cannot be compared, only their duration.
* After the dialer change on Thursday, unanswered legs are not logged, so per-dial shares after the change are overstated.
* Departure dates were reconstructed, because the CRM does not keep deactivation history.
* The team is small, so one person visibly moves the weekly totals.
* Business results are not disclosed (NDA), and the window is too short.

== Exercises ==
# Why is claim no. 2 correct as a daily figure but not as a weekly one? ''(Answer: 269 is Monday and 1,055 is Thursday across all accounts, so both numbers are single days. A weekly claim would need weekly totals.)''
# The result &quot;+80%&quot; becomes &quot;+33%&quot;. What changed in the calculation? ''(Answer: +80% excludes the live supervisor's calls. +33% includes them. The supervisor's account made several hundred dials a week before the agent and almost none after.)''
# A report says &quot;answer rate rose from 10% to 20%&quot; and names no denominator or period. Which verdict fits? ''(Answer: not checkable, as with claim no. 5, because the metric is not fully defined.)''
# What must a weekly report of an AI supervisor include so that it can be recomputed? ''(Answer: the formula, the denominator and the period for every number, as the cases of claims 7 and 8 show.)''

== Source ==
Technical report, 2026, a case under a non-disclosure agreement: {{doi|10.5281/zenodo.23926851}}

== See also ==
* [[Artificial intelligence/AI content engine economics]]</text>
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      <text bytes="8502" sha1="6q0pxf0yr47l6hlp5yds4yyvciww8qn" xml:space="preserve">{{Short description|Checking an AI supervisor's weekly report}}
This learning resource is a sub-page of [[Artificial intelligence]].

When an AI agent leads a team and writes a report to the people above it, how can a reader check that the report is correct? This resource teaches one method: recompute every claim in the report from the raw data. It uses one measured case, with its limits stated openly.

{{RightTOC}}

== Learning objectives ==
After working through this resource, you should be able to:
* define an AI supervisor and describe what its daily work includes;
* explain why a before-and-after comparison cannot isolate the effect of an agent when several things change at once;
* recompute a report claim from raw data and assign it one of four verdicts;
* list what a weekly report must contain so that it can be recomputed;
* name the limitations of a small before-and-after case.

== 1. What an AI supervisor is ==
An AI supervisor is a software agent that does the daily work of a team lead. It watches the team's numbers, reviews calls, flags problems, answers team questions and reports upward.

== 2. The measured case ==
An autonomous Claude agent led a small remote phone team for three weeks. The team had 4–8 people who call partners for a telehealth company serving the US market. People stayed in the loop: a live team supervisor, a head of department and senior management. The company is under a non-disclosure agreement, so only the numbers and the way they were calculated are published. Raw data is not published.

The agent had access to 584 chat messages, 272 of them addressed to the agent. Over 16 days the agent's activity looked like this:

{| class=&quot;wikitable&quot;
! Type of model run !! Count
|-
| All model runs || 459
|-
| Runs triggered by chat messages (approximately) || 279
|-
| &quot;Floor&quot; checks (from the end of week 2) || 72
|-
| Structured interview of the live supervisor about the auto-dialer (week 2) || 44
|-
| Daily &quot;pulse&quot; || 31
|-
| Nightly reviews || 13
|-
| Morning plans || 11
|-
| Interview briefs || 9
|}

An auto-dialer is a system that dials numbers automatically; it is not a person. On the Monday of week 3, 80 of 84 runs did no work.

In week 3, talk time with partners was 33–80% above the best week before the agent. The figure is 80% if the live supervisor's calls are excluded and 33% if they are included. Before the agent, the supervisor's account made several hundred dials a week; afterwards it made almost none. The first two weeks with the agent were worse than before.

== 3. Before and with the agent: why the comparison is hard ==
Several things changed in week 3 at the same time, so the effect of the agent cannot be separated:
* The auto-dialer was brought back. The share of auto-dialed calls went from 5% in week 2 to 84% in week 3.
* The number of active callers grew from 4 to 6.
* The call filter was switched off on one caller's phone.
* The dialer's connection order was changed.
* Rules were introduced against redialing the same number within 3 hours.
* The agent did almost no work for most of Monday.

== 4. Who found the defects and who decided about people ==
Both main breakdowns of the period were found by people, not by the agent: the call filter on a caller's phone and the dialer's connection order.

Five callers left during the three weeks. People let all of them go: management, the live supervisor and the head of department. The agent let no one go, and its logs contain 0 recommendations to dismiss anyone. Because the agent recommended nothing, its rule &quot;recommend, but do not decide&quot; was never tested.

In week 3 the head of department addressed most of their messages to the agent. They asked it to find promised callbacks that had not been made, using the calls. The agent found 8.

== 5. Auditing the weekly report ==
At the end of week 3 the agent's weekly report contained 9 checkable claims. Each was recomputed from the raw sources.

{| class=&quot;wikitable&quot;
! # !! Claim in the report !! Result of recomputation !! Verdict
|-
| 1 || &quot;Live conversation up 41%&quot; || No calculation exists in the agent's files. Nearest figures: answered calls of 60 s or more +40% against week −2; talk time +80% against week −2 and +166% against week 2. Not reproducible. || Not reproducible
|-
| 2 || &quot;Calls nearly quadrupled, 269 → 1,057&quot; || This is a daily figure, not a weekly one: Monday 269 → Thursday 1,055 across all accounts. Correct as a daily figure. || True as written
|-
| 3 || &quot;4 people let go; all operator accounts cleaned up&quot; || 4 departures in week 3 is correct, but access removal was not finished at the time of the check. || Partly false
|-
| 4 || &quot;A new caller is the team's best at conversation today&quot; || On Friday this caller had the most long (60 s or more) answered calls, but not the most talk time. || True under one counting
|-
| 5 || &quot;A second new caller shows good results&quot; || No metric is named. || Not checkable
|-
| 6 || &quot;Three candidates start trial days; all scored 90/100 in the AI interview&quot; || The score is a fixed label stored for the verdict, not a measurement. || Misleading
|-
| 7 || &quot;Call screening: 45 of 51 calls ended in nothing; after the fix the result doubled&quot; || After processing all transcripts: 47 of 54. The share of calls of 60 s or more rose ×1.8. Long calls per hour were unchanged. || True under one counting
|-
| 8 || &quot;Dialer fix: real conversations 8% → 15%&quot; || Reproducible only if the denominator is all dials, &quot;before&quot; is Wednesday and Thursday morning, and &quot;after&quot; is Friday. Tuesday was 17%. || True under one counting
|-
| 9 || &quot;N partners had no owner at the start of the week; far fewer now&quot; || The starting figure is in no table and does not match the assignment log. || Not reproducible
|}

Summary: 1 claim is true as written (no. 2); 3 are true only under one way of counting (nos. 4, 7, 8); 2 are partly false or misleading (nos. 3, 6); 3 are not reproducible or not checkable (nos. 1, 5, 9).

[[File:AI supervisor weekly report audit - nine claims by verdict (2026).svg|thumb|600px|center|The nine checkable claims by verdict.]]

The source study concludes that an AI manager needs a named human checker who recomputes its report.

== 6. Four verdicts as a method ==
To apply the method to your own data:
# Split the report into single claims, each with one number or one statement.
# For each claim, find the raw source and recompute it yourself.
# Assign one verdict: true as written; true only under one way of counting; partly false or misleading; not reproducible or not checkable.
# For every &quot;one way of counting&quot; verdict, write down the way: the denominator, the period and the metric.

== 7. Limitations ==
* There is no control group, only before and after.
* Week 1 was a transition. In week 2 the team shrank to 4 callers. Week 3 covers only Monday to Friday.
* Call classification exists only for week 3 and was not checked against human labelling (coverage of answered calls 77–92%). Therefore the quality of conversations before the agent cannot be compared, only their duration.
* After the dialer change on Thursday, unanswered legs are not logged, so per-dial shares after the change are overstated.
* Departure dates were reconstructed, because the CRM does not keep deactivation history.
* The team is small, so one person visibly moves the weekly totals.
* Business results are not disclosed (NDA), and the window is too short.

== Exercises ==
# Why is claim no. 2 correct as a daily figure but not as a weekly one? ''(Answer: 269 is Monday and 1,055 is Thursday across all accounts, so both numbers are single days. A weekly claim would need weekly totals.)''
# The result &quot;+80%&quot; becomes &quot;+33%&quot;. What changed in the calculation? ''(Answer: +80% excludes the live supervisor's calls. +33% includes them. The supervisor's account made several hundred dials a week before the agent and almost none after.)''
# A report says &quot;answer rate rose from 10% to 20%&quot; and names no denominator or period. Which verdict fits? ''(Answer: not checkable, as with claim no. 5, because the metric is not fully defined.)''
# What must a weekly report of an AI supervisor include so that it can be recomputed? ''(Answer: the formula, the denominator and the period for every number, as the cases of claims 7 and 8 show.)''

== Source ==
Technical report, 2026, a case under a non-disclosure agreement: {{doi|10.5281/zenodo.23926851}}

== See also ==
* [[Artificial intelligence/AI content engine economics]]

{{BookCat}}</text>
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  <page>
    <title>Liouville theory with a central charge less than one</title>
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      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
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      <comment>Structure</comment>
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== Features of correlation functions ==

== Attempts at a construction ==

== Living without a construction == 

=== Proof of existence ===

=== Minimal strings === 


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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    <revision>
      <id>2844768</id>
      <parentid>2844765</parentid>
      <timestamp>2026-11-06T22:15:12Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
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      <comment>Lead section</comment>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, but cannot prove the consistency of the theory – only test it numerically.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) No theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

== Attempts at a construction ==

== Living without a construction == 

=== Proof of existence ===

=== Minimal strings === 

== References ==

{{Reflist|refs=
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

== Living without a construction == 

=== Proof of existence ===

=== Minimal strings === 

== References ==

{{Reflist|refs=
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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      <comment>Another section</comment>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

=== Proof of existence ===

=== Minimal strings === 

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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      <id>2844778</id>
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      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

=== Proof of existence ===

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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        <username>Sylvain Ribault</username>
        <id>2127778</id>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

=== Proof of existence ===

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;
An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

=== Path integral construction ===

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

=== Reproducing known features ===

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;

=== Specific attempts === 

An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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        <id>2127778</id>
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      <comment>/* Application to the Virasoro minimal string */</comment>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

=== Path integral construction ===

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

=== Reproducing known features ===

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;

=== Specific attempts === 

An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

A recent application of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is the construction of the Virasoro minimal string. This application did not need more than the bootstrap definition of the theory, including for finding a matrix integral description of the resulting string theory. Here again, a construction might have been convenient, but it was possible to do without it. In fact, when it came to deducing correlation numbers of the string theory from correlation functions of the CFT, it was essential to know their analytic properties extensively: a construction that would come with limitations on the allowed values of momentums might not have been very helpful.

== References ==

{{Reflist|refs=
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

=== Path integral construction ===

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

=== Reproducing known features ===

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;

=== Specific attempts === 

An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

A recent application of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is the construction of the Virasoro minimal string.&lt;ref name=&quot;cemr23&quot;/&gt; This application did not need more than the bootstrap definition of the theory, including for finding a matrix integral description of the resulting string theory. A construction would have been convenient, but it was possible to do without it. 

When it came to deducing correlation numbers of the string theory from correlation functions of the CFT, it was essential to know their analytic properties extensively:&lt;ref name=&quot;kl25&quot;/&gt; a construction that would come with limitations on the allowed values of momentums might not have been very helpful.

== References ==

{{Reflist|refs=
&lt;ref name=&quot;kl25&quot;&gt;{{cite web | last1=Khromov | first1=Dmitry | last2=Litvinov | first2=Alexey | title=On correlation numbers $V_{0,4}$ and $V_{1,1}$ in Virasoro Minimal String Theory | website=arXiv.org | date=2025-09-30 | url=https://arxiv.org/abs/2509.25960 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;cemr23&quot;&gt;{{cite journal | last1=Collier | first1=Scott | last2=Eberhardt | first2=Lorenz | last3=Mühlmann | first3=Beatrix | last4=Rodriguez | first4=Victor A. | title=The Virasoro minimal string | journal=SciPost Physics | publisher=Stichting SciPost | volume=16 | issue=2 | date=2024-02-26 | issn=2542-4653 | doi=10.21468/scipostphys.16.2.057 | doi-access=free | url=https://arxiv.org/abs/2309.10846}}&lt;/ref&gt;
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution. We then 
discuss the attempts at constructing &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory. Can they reproduce the known features of the theory, or do they describe something else? 

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To really understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Attempts at a construction ==

=== Path integral construction ===

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

=== Reproducing known features ===

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;

=== Specific attempt === 

An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the bootstrap construction.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

A recent application of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is the construction of the Virasoro minimal string.&lt;ref name=&quot;cemr23&quot;/&gt; This application did not need more than the bootstrap definition of the theory, including for finding a matrix integral description of the resulting string theory. A construction would have been convenient, but it was possible to do without it. 

When it came to deducing correlation numbers of the string theory from correlation functions of the CFT, it was essential to know their analytic properties extensively:&lt;ref name=&quot;kl25&quot;/&gt; a construction that would come with limitations on the allowed values of momentums might not have been very helpful.

== References ==

{{Reflist|refs=
&lt;ref name=&quot;kl25&quot;&gt;{{cite web | last1=Khromov | first1=Dmitry | last2=Litvinov | first2=Alexey | title=On correlation numbers $V_{0,4}$ and $V_{1,1}$ in Virasoro Minimal String Theory | website=arXiv.org | date=2025-09-30 | url=https://arxiv.org/abs/2509.25960 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;cemr23&quot;&gt;{{cite journal | last1=Collier | first1=Scott | last2=Eberhardt | first2=Lorenz | last3=Mühlmann | first3=Beatrix | last4=Rodriguez | first4=Victor A. | title=The Virasoro minimal string | journal=SciPost Physics | publisher=Stichting SciPost | volume=16 | issue=2 | date=2024-02-26 | issn=2542-4653 | doi=10.21468/scipostphys.16.2.057 | doi-access=free | url=https://arxiv.org/abs/2309.10846}}&lt;/ref&gt;
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


[[Category:Research]]
[[Category:Conformal field theory]]</text>
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== 2026-11-07 ==
[[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.&lt;!-- Glow-vandalism1 @ 1794023792631.4s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:56, 7 November 2026 (UTC)</text>
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    <title>Tigre Language</title>
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      <comment>Created main page for Tigre Language</comment>
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      <text bytes="457" sha1="s980i326qd5brprgn4my0q8uy57ip5j" xml:space="preserve">= Tigre Language =

Welcome to learn Tigre (ትግሬ) language.

Tigre is a Semitic language spoken in Eritrea and Sudan.
These lessons are free for everyone.
English - Arabic - Tigre.

== Lessons ==

* [[Tigre Language - My Day]]
* [[Tigre Language - My Work Day]]

== How to learn ==

1. Read the English sentence.
2. Read the Arabic translation.
3. Read and repeat the Tigre sentence.
4. Try to write a new sentence like it.

[[Category:Tigre language]]</text>
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      <text bytes="282" sha1="8q1ru27agb920xnhgvugy1rne5b85c4" xml:space="preserve">== Summary ==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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      <comment>Hyperstrateji: Bir firmanın ürün, fiyat, algı ve ekosistem katmanlarını eşzamanlı olarak yöneterek pazarın kurallarını kendi lehine yeniden yazma sanatı.</comment>
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